Let \(G\) be a group of order \(s^2t\), \(s,t>1\), \(J= \{A_i\mid 0\leq i\leq t\}\) be a family of \(t+1\) subgroups of order \(s\) of \(G\). For any \(A_i\in J\) let \(A^*_i\) be a subgroup of \(G\) of order \(st\), containing \(A_i\); \(J^*= \{A^*_i\mid 0\leq i\leq t\}\). A pair \((J,J^*)\) is called 4-gonal family (Kantor family) of type \((s,t)\), if the following holds: (K1) \(A_iA_j\cap A_k=1\) for any pairwise different \(i,j,k\); (K2) \(A^*_i \cap A_j =1\) for \(i\neq j\). W. M. Kantor has shown that the existence of elation generalized quadrangles is equivalent to the existence of a 4-gonal family of subgroups of the elation group \(G\). Let \({\mathcal S} =J\cap J^*\). A nontrivial subgroup \(X\) of \(G\) is called \({\mathcal S}\)-factor of \(G\), if the following holds: \[ (U\cap X)(V\cap X)=X\quad \text{ for all } U,V\in{\mathcal S}\text{ satisfying }UV=G.\tag{F} \] Let \(X\) be an \(\mathcal S\)-factor of \(G\), \(J_X=J\cap X\) and so on. Then there exist integers \(\sigma \geq 1\) and \(\tau \geq 1\) such that \(|X|=\sigma^2\tau\), \(|A\cap X|=\sigma\) and \(|A^*\cap X|=\tau\) for all \(A\in J\). Moreover, either \(\sigma=1\) and \(X\) is a subgroup of \(\bigcap_{A\in J}A^*\), or \(\sigma>1\), \(\tau=t\) and \((J_X,J_X^*)\) is a Kantor family in \(X\) of type \((\sigma,\tau)\). If \(X\) is a normal subgroup in \(G\) and has type \((1,t)\), then the Kantor GQ is a skew translation generalized quadrangle (STGQ) with \(X\) being a full group of symmetries about the base point. The main result of this paper is Theorem 2.5. Let \(G\) be a group of order \(s^2t\) admitting a Kantor family \((J, J^*)\) of type \((s,t)\) and a normal \(\mathcal S\)-factor \(X\) of type \((\sigma,t)\). Then either \(G\) is a group of prime power order, or \(\sigma>1\), \(|G|\) has exactly two prime divisors and \(X\) is a Sylow subgroup of \(G\). As a corollary, the parameters of STGQ are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.