Let $G$ be a finite group.
Let $n$ be an integer such that $G$ has the following property:
"$n$-Free property": $G$ can be generated by $n$ elements, and for any two generating $n$-tuples $(g_1,\ldots,g_n)$, $(g_1',\ldots,g_n')$ there is an automorphism $\sigma$ of $G$ such that $\sigma(g_i) = g_i'$.
Under this assumption, if $A,B$ are two quotients of $G$, then is every subgroup of $A\times B$ which projects onto $A$ and $B$ which is also generated by $n$ elements also a quotient of $G$?
Essentially, the idea behind my question is: Is the "Free property" sufficient to say that $G$ is a free pro-$C$ group of rank $n$ (in the sense of Ribes-Zalesski's characterization of free profinite groups in their book "Profinite Groups"), where here $C$ is the class of all finite subdirect products of quotients of $G$.