Suppose we have a group $G = H \rtimes K$, where $K = \langle \alpha \rangle$ is of order 2. We suppose that $K$ centralizes some 2-Sylow subgroup. Then every involution is of one of these forms: 1) $h \in H$, 2) $k$ a conjugate of $\alpha$, 3) $hk$ where $h$ is an involution in $H$ and $k$ is a conjugate of $\alpha$ centralizing it.
We want to count the number of involutions of type 3). Is there a way to do that if we know the number of involutions of type 1) and 2)? What about the number of conjugacy classes of involutions of type 3) if we know that all those of type 1) are conjugate? (Those of type 2) are conjugate by hypothesis).