The prime-power wreath-product strategy conjecture for involution-generated groups

Let $G$ be a finite group generated by involutions, and let $H$ be a $2$-group acting faithfully on the set of switches. The wreath product is denoted by $G\wr H$. Prime-power wreath-product strategy conjecture. There exists a surjective strategy for $G\wr H$. This would extend the known construction for $G\wr C_2$ to adversarial groups of order a power of $2$; the source leaves the broader construction open.