Finite \(2\)-groups with small centralizer of an involution. II (Q5958871)

In part I of the paper [J. Algebra 241, No. 2, 818-826 (2001; Zbl 0988.20009)], the author classified the finite 2-groups \(G\) with the property that \(G\) possesses an involution \(t\) such that \(C_G(t)=\langle t\rangle\times C\), where \(C\) is a cyclic group. In the present paper, the author classifies the finite 2-groups \(G\) which possess an involution \(t\) such that \(C_G(t)=\langle t\rangle\times Q\), where \(Q\) is a (generalized) quaternion group. As a direct application of this result, it is obtained a classification of finite 2-groups which have more than three involutions but which do not have an elementary Abelian subgroup of order 8. Such groups have been considered by \textit{D. J. Rusin} [J. Algebra 149, No. 1, 1-31 (1992; Zbl 0795.20008)] by using a heavy cohomological machinery.