Identity functor controls strong fusion for saturated fusion system on abelian group

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., abelian group of prime power order) must also satisfy the second group property (i.e., resistant group of prime power order)
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Statement

In terms of control of strong fusion

Suppose $P$ is an abelian group of prime power order. Suppose $\mathcal{F}$ is a saturated fusion system on $P$. Then, the identity functor (i.e., the functor sending a group to itself) is a conjugacy functor that controls strong fusion on $P$.

In the language of resistant

Any abelian group of prime power order is a resistant group of prime power order.

Related facts

Group theory version

• Identity functor controls strong fusion for abelian Sylow subgroup