Characteristic p-functor that gives a characteristic subgroup

Definition
Suppose $$p$$ is a prime number and $$G$$ is a finite group such that $$W$$ is a conjugacy functor for $$G$$ for the prime $$p$$ arising from a characteristic p-functor. We say that $$W$$ is a characteristic p-functor that gives a characteristic subgroup if it satisfies the following equivalent conditions:


 * 1) For every pair of $$p$$-Sylow subgroups $$P,Q$$ of $$G$$, $$W(P) = W(Q)$$.
 * 2) For every pair of $$p$$-Sylow subgroups $$P,Q$$ of $$G$$, $$W(P)$$ is a normal subgroup of $$Q$$.
 * 3) Each of these:
 * 4) * $$W$$ is a weakly closed conjugacy functor and there exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that $$W(P) \le O_p(G)$$ where $$O_p(G)$$ is the p-core of $$G$$.
 * 5) * $$W$$ is a weakly closed conjugacy functor and for every $$p$$-Sylow subgroup $$P$$ of $$G$$, $$W(P) \le O_p(G)$$ where $$O_p(G)$$ is the $$p$$-core of $$G$$.
 * 6) Each of these:
 * 7) * There exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that $$W(P)$$ is a characteristic subgroup of $$G$$.
 * 8) * For every $$p$$-Sylow subgroup $$P$$ of $$G$$, $$W(P)$$ is a characteristic subgroup of $$G$$.
 * 9) Each of these:
 * 10) * There exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that $$W(P)$$ is a normal subgroup of $$G$$.
 * 11) * For every $$p$$-Sylow subgroup $$P$$ of $$G$$, $$W(P)$$ is a normal subgroup of $$G$$.