I-automorphism-invariance satisfies intermediate subalgebra condition

Statement

Suppose ${\mathcal {V}}$ is a variety of algebras, $A$ is an algebra of ${\mathcal {V}}$ and $B$ is a subalgebra of $A$ such that for every I-automorphism (?) $\varphi$ of $A$, $\varphi (B)\subseteq B$. Then, if $C$ is a subalgebra of $A$ containing $B$, every I-automorphism of $C$ also sends $B$ to itself.

Related facts

Particular cases

• Normality satisfies intermediate subgroup condition: For the variety of groups, the I-automorphisms are the inner automorphisms, and the subgroups invariant under these automorphisms are the normal subgroups. Thus, the statement translates to saying that a normal subgroup of a group is normal in every intermediate subgroup.

Other related facts

• Ideal property satisfies intermediate subalgebra condition