Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant

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Suppose H is a subgroup of a group G satisfying the following two conditions:

  1. H is a Fully normalized subgroup (?) of G, i.e., every automorphism of H extends to an inner automorphism of G.
  2. H is a Potentially fully invariant subgroup (?) of G, i.e., there exists a group K containing G in which H is a fully invariant subgroup.

Then, H is a Centralizer-annihilating endomorphism-invariant subgroup (?) of G: For every endomorphism \sigma of G whose kernel contains C_G(H), \sigma(H) \le H.

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