In [pg. 24](https://books.google.com/books?id=5lv_CAAAQBAJ&pg=PA24) of his book on Galois cohomology, Serre gives the following exercise: 


"Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties:  

(i) $P$ is a pro p-group  

(ii) $G$ is finite.  

(iii) A Sylow p-subgroup of $G$ lifts to $E.$  

(iv) $G$ does not lift to $E.$" 


I am not sure how this can happen, allow me to explain. Let us start by assuming that all groups involved are finite. Suppose that $S$ is a Sylow $p$-subgroup of $G$ and that it lifts to $E.$ Taking the fiber product along the inclusion $S \to G,$ we get the exact sequence 
$$1 \rightarrow P \rightarrow E \times\_G S \to S \rightarrow 1.$$ By the assumption that $S$ lifts to $E,$ this exact sequence is split.  

If $P$ is an abelian $p$-group, this should imply that there is a lift of $G$ to $E,$ since the map $res:H^2(G,P) \to H^2(S,P)$ is injective by the assumption that $S$ is a Sylow $p$-subgroup. 


Now, it seems to me that using the above argument for when $P$ is an abelian $p$-group, we should be able to do some devissage argument to show that given any extension $1 \to P \to E \to G \to 1$ of profinite groups and a lift of a Sylow $p$-subgroup of $G$ to $E$ as in the exercise, there is a lift of $G$ to $E$ as well. But this is in contradiction to the exercise. 


Obviously, this means I have made a mistake in my thinking, but I have spent some time on it but can not seem to find it. Could someone help me finding where this mistake is?


**Update**
I now see where my mistake lies. I will post this as an answer tomorrow.



No lifts in an exact sequence of profinite groups?
Here is the problem with my conjectural argument. 


My hope that one could reduce this to the case where $P$ is finite abelian, but this does not work. This reduction can however be made if one further assumes that $H^1(G,P) \to H^1(S,P)$ is an isomorphism for any abelian $p$-group. To be able to reduce the general case, it turns out that one need to be able to extend the given embedding $s: S \rightarrow E$ to an embedding $G \to E.$ This can be done under the given assumptions and then the devissage argument works.


