Reddit mentions of Car Bluetooth Receiver Stereo AUX Adapter with Handsfree Calling by GOgroove - SMARTmini AUX - Portable 3.5mm Wireless Music A2DP Adapter Dongle with 30-Foot Streaming for Smartphones & Tablets

If your radio doesn't have bluetooth input, you can get a bluetooth to aux device. https://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G

I would look into this product (if it suits your needs). I bought a different one from them for my car (as I don't have a 3.5mm plug) and it's been flawless for a year.

http://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G/ref=sr_1_4?s=electronics&ie=UTF8&qid=1421681246&sr=1-4&keywords=gogroove&pebp=1421681248997&peasin=B008XMV05G

If you're gonna go the bluetooth route, check out this one too:

http://www.amazon.com/gp/product/B008XMV05G/

It's cord free if you can charge it once a week. Or you can leave it constantly plugged in if you prefer. Looks cleaner and works well.

I just ordered this one $20 cheaper and does calls and a2dp.

With this one I can hide the wires and have it on my dashboard. Ideally I wanted something I can connect directly to the car's 12v so I don't have to worry about plugging it in and out. This one has a battery and is charged through a usb-3.5mm adapter. I'll still have to power it on when getting in and out but for $29.99 I'll deal with it.

Also bonus to this one, I can unplug it from it's little mount and plug it into any 3.5mm jack (say at a friends house or party) and not have to worry about leaving my phone plugged into the stereo unattended.

maybe? http://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G

My set up for my older model Jeep:

Mount is here:
http://www.amazon.com/gp/product/B00C7B2NV4/ref=oh_aui_detailpage_o04_s00?ie=UTF8&psc=1

Charger is here:
http://www.amazon.com/gp/product/B00M9TBTDM/ref=oh_aui_detailpage_o03_s01?ie=UTF8&psc=1

Smaller X-grip barely fits the nexus 6 with no case, and case and you need the bigger one for Phablets. THe wirelss charger needs to be mounted on the lower end of the grip for it to pick up well.

Seems to work beautifully so far.

I have used a similar set up with my MotoX mounted here for the last year, but without the wireless charger. Oh yeah, and I also have a bluetooth audio/phone adapter mounted on the dash, which you can see on the first picture:
http://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G/ref=sr_1_1?s=wireless&ie=UTF8&qid=1418506201&sr=1-1&keywords=bluetooth+car+speakerphone+smart+mini

Transmitting FM radio takes special hardware that phones don't have. A very few older phones can receive FM radio, but that's it. I highly doubt you can find exactly what you described, but there are alternatives.

There are cigarette lighter Bluetooth radio transmitters, where you can use your phone to connect by Bluetooth, and the device converts Bluetooth to FM radio. You don't need a special app for that. https://www.amazon.com/dp/B06X9K9X1G/ref=cm_sw_r_cp_apa_P-F0AbS877RMS

If your car has an auxiliary audio jack, there are lots of Bluetooth receivers for that, and the sound quality is much better. I've bought 3 of these, for various purposes including my cars: https://www.amazon.com/dp/B008XMV05G?ref=yo_pop_ma_swf

There are Bluetooth to aux adapters like this that might serve your purpose.

Still a stupid move on Apple's part.

I prefer this one. It's dirt cheap and does exactly what I want such as streaming Google Play Music and the occasional phone call.

http://www.amazon.com/gp/aw/d/B008XMV05G/ref=mp_s_a_1_7?qid=1416782429&sr=8-7&pi=AC_SX110_SY165_QL70

I bought this bad boy for my wife's car and paired it with [this set of bluetooth buttons] ( http://www.amazon.com/gp/aw/d/B00KPZN5UC/ref=mp_s_a_1_1?qid=1416782490&sr=8-1&pi=AC_SY200_QL40) . The pair together are nearly as good as having it natively in your car.

http://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G This is what you're looking for.

https://www.amazon.com/dp/B008XMV05G/ref=cm_sw_r_awd_oYVNub0NHB8BT and there are plenty of cheaper knock offs on ebay.

Okay, I guess I was not hallucinating: https://www.amazon.com/Bluetooth-Receiver-Adapter-Handsfree-GOgroove/dp/B008XMV05G

Somewhat outside the box suggestion but if you've got an aux port in your car you could just do this instead....

http://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G/ref=sr_1_1?s=wireless&ie=UTF8&qid=1412831961&sr=1-1&keywords=go+groove+smart+mini

You can add bluetooth to any car for about $20 with something like this: https://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G

So the Soundbot turns on with the car? I have a GoGroove SMARTMini AUX and the one annoyance I have with it is the fact that I have to manually turn it on whenever I get into my car. First world problems amirite

https://www.amazon.com/GOgroove-SMARTmini-Bluetooth-Hands-Free-Streaming/dp/B008XMV05G

Problem Solved!