Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal

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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Automorph-join-closed subnormal subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Conjugate-join-closed subnormal subgroup (?))
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Statement

Statement with symbols

If H is an automorph-join-closed subnormal subgroup of K and K is a normal subgroup of G, then H is a conjugate-join-closed subnormal subgroup of G.

Related facts

Proof

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