A stereo headset with a mic must have 4 poles: left earpiece, right earpiece, mic and ground. And that is also the case with the Logitech H111.
With a three-pin plug, it's probably not possible. This would most likely be a stereo headset WITHOUT a micro (left, right, earth). It could also be a mono headset WITH a micro, but this is hardly ever the case nowadays.

Stereo is not needed for a telephone voice connection, not even for the Fritzfon. But the critical point is the contact assignment for the micro. Unfortunately, there are 2 variants for the 4-pole things, namely

Variant 1:
Ground on the second rearmost contact ring (i.e. L, R, Gnd, Mic as seen from the plug tip; quite common variant, e.g. Samsung, etc.).

Variant 2:
Ground on the rearmost contact ring (i.e. L, R, Mic, Gnd as seen from the plug tip; nowadays the rare variant).


AVM explains under the following link what the C5 needs:
https://avm.de/service...

So variant 1, and thus the H111 should fit.

By the way, Delock has a clever adapter cable for switching between the above variants 1 and 2, simply plug it in where necessary:
Delock item no. 62498
..... and it's even available from Digitec: Art. no. 5741713

A stereo headset with microphone is always 3-pole:

1 pole left
1 pole right
1 pole the microphone
has 3 rings on the plug and is thanks to the 3.5 mm jack plug exactly what you are looking for your Fritzfon C5.