It is well known that any group is a quotient a free group by a normal subgroup that is free. More precisely if $G$ is a group the exists a short exact sequence of groups $$1\rightarrow F^{'}\rightarrow F\rightarrow G\rightarrow 1 $$ where $F^{'}$ and $F$ are free groups.

Q1: Is any profinite group a quotient of a free profinite group by a normal subgroup that is free profinite?

Assuming that the answer to the first question is negative

Q2: what can we say about a profinite group if initially we know that it is a quotient of a free profinite group by a normal subgroup that is free profinite?

By the second question I do mean if such profinite group has some cohomological properties.

Normal subgroup of free profinite groups1978 AMS, Mathematics of the USSR-Izvestiya, 12(1) (iopscience.iop.org/article/10.1070/IM1978v012n01ABEH002289). It addresses the problem of understanding which normal subgroups of free profinite groups are free profinite. $\endgroup$ – YCor Nov 15 '17 at 13:17