Left-inner right-monoidal subgroup property

Statement
A subgroup property $$p$$ is termed a left-inner right-monoidal subgroup property if it can be expressed using a function restriction expression of the form:

Inner automorphism $$\to b$$

where $$b$$ is a monoidal function property -- in other words, for any group, the collection of functions from the group to itself satisfying property $$b$$ forms a monoid under composition.

Weaker metaproperties

 * Stronger than::Upper join-closed subgroup property:
 * Stronger than::Left-inner subgroup property
 * Stronger than::Left-extensibility-stable subgroup property
 * Stronger than::Intermediate subgroup condition