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 fact about::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.

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