In GAP small group library, The group $$[32,2]=\langle a,b,c\mid a^4=b^4=c^2=1, ba=abc, [a,c]=[b,c]=1 \rangle.$$
We say $G/N$ is a maximal quotient group if there exists no quotient group $G/K$ such that $G/N$ is isomorphic to a quotient group of $G/K$.
Use GAP, we found that $[32,2]\times C_2$ has the maximal nonabelian quotient group which is isomorphic to $[32,2], [32,22]$ or $[32,23]$; the group $[32,2]\times C_2\times C_2$ has the maximal nonabelian quotient group which is isomorphic to $[32,2]\times C_2, [32,22]\times C_2$ or $[32,23]\times C_2$. Define $n$-copy direct product of $C_2$ by $C_2^n$.
I have following Question: is it possible to prove that if $H$ is a maximal quotient group of $[32,2]\times C_2^n$, then $H$ is isomorphic to $[32,2]\times C_2^{n-1}, [32,22]\times C_2^{n-1}$ or $[32,23]\times C_2^{n-1}$.
Thanks for your time.