Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd\_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free pro-$p$ group of rank $d(G)$. Must $L$ be free pro-$p$ ?


Here, $d(G)$ is the smallest cardinality of a generating set of $G$. 



An epimorphism into a profinite group
No.


Let $G = D\_p \times F$,
where $D\_p = \langle a,b \mid\ a^p = 1,\ b^2 = 1,\ a^b \ (:=b^{-1}ab)=a^{-1}\rangle$
is the dihedral group of order $2p$
and $F$ is the free pro-$p$ group on two generators, say, $x,y$.
Then $L = \langle a \rangle \times F$ is of index $2$ in $G$, pro-$p$, but not free,
and there is a surjection $G \to F$.
Since $H := \langle ax, by \rangle$ is clearly of rank $2$,
it suffices to show that $H = G$, i.e., that $a,b,x,y \in H$.


As $b,y$ are of coprime (supernatural) orders and commute,
$\langle b y \rangle = \langle b \rangle \times \langle y \rangle$,
so $b, y \in H$.
Thus $x a^{-1} = a^{-1} x = (ax)^b \in H$.
Therefore $x^2 = (x a^{-1}) (a x) \in H$
and $a^2 = (a x) (a^{-1} x)^{-1} \in H$.
As $2$ is prime to the orders of $x$ and $a$,
we have $\langle x^2 \rangle = \langle x \rangle$
and $\langle a^2 \rangle = \langle a \rangle$,
so $x, a \in H$.


