If $A\lhd B\lhd C$ with $[C:A]=p^n$, is there $D\lhd A$ with $D\subset C$ and $D/A$ a finite $p$-group?

Question

Let $A$ be a (possibly infinite) group.

Consider subgroups $C\lhd B\lhd A$, and assume that $A/B$ and $B/C$ are both finite $p$-groups.

Is there necessarily a subgroup $D$ normal in $A$ and contained in $C$ such that $A/D$ is a finite $p$-group?

This is related to another question:

If the subgroup $H$ of $G$ is open in pro-$p$ topology, does it inherit the pro-$p$ topology?

Answer 1

Since the core of $C$ in $A$ has finite index in $A$, we can assume without loss of generality that $A$ is finite.

But then we can just choose $D=O^p(A)$, which is the smallest normal subgroup of $A$ such that the quotient is a $p$-group or, alternatively, the subgroup of $A$ generated by the $p'$-elements of $A$.

So the answer is yes.