Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Intermediately characteristic subgroup (?) and Transfer-closed characteristic subgroup (?)), to another known subgroup property (i.e., Intermediately characteristic subgroup (?))
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Statement

Property-theoretic statement

If $*$ denotes the composition operator, we have:

Intermediately characteristic $*$ transfer-closed characteristic $\leq$ Intermediately characteristic

Verbal statement

Any intermediately characteristic subgroup of a transfer-closed characteristic subgroup is intermediately characteristic.

Facts used

1. Characteristicity is transitive
2. Subhomomorphism relation between transfer-closure operator and intermediately operator over composition operator