Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ containing $A$?

The existence of some basis is guaranteed since $H$ is a free group (Nielsen-Schreier Theorem). Even the case when $A$ is a singleton is not clear to me. On the other hand, It may well be that the there is no need to assume that the index is finite, and just take $H$ to be finitely generated. Furthermore, I find the following profinite analogue interesting too:

Let $F$ be a free profinite group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq_o F$. Must there be a basis (free topological generating set) for $H$ containing $A$?