Let $G$ be a finite solvable group of order $p^2q^2$, where $p>q$ and $q\nmid p-1$. Let $G$ has the following presentation:
$\langle a , b ,c \vert a^p=b^p=c^q=1, ab=ba, cac^{-1}=a^{i}b^{j}, cbc^{-1}=a^{k}b^{l}\rangle$
and $ {\left( {\begin{array}{*{20}c} i & j \\ k & l \end{array}}\right)} $ has order $q^2$ in $GL(2,p)$. Is it possible to classify such groups with $O_q(G)=1$? (Recall that $O_q(G)=\cap_{g\in G} Q^g$, where $Q\in {\rm Syl}_q(G)$ ). Many thanks for your thoughts on this!
There is an old paper "Classification of groups of order $p^2q^2$" by G.Cheissin where all such groups were described (unfortunately, in Russian). See http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3913&option_lang=eng