Involution fusion pattern

Definition with symbols
Two groups $$G$$ and $$G^*$$ are said to have the same involution fusion pattern if there is an isomorphism $$x \mapsto x^*$$ from a 2-Sylow subgroup $$S$$ of $$G$$ to a 2-Sylow subgroup $$S^*$$ of $$G^*$$ such that $$x,y \in S$$ are conjugate in $$G$$ if and only if $$x^*, y^*$$ are conjugate elements in $$S^*$$.

Stronger equivalence relations

 * Centralizer of involution pattern