Characteristicity is transitive for any variety of algebras

Statement
Suppose $$\mathcal{V}$$ is a variety of algebras. Suppose $$A$$ is an algebra of $$\mathcal{V}$$, $$B$$ is a subalgebra of $$A$$ that is characteristic in $$A$$ (i.e., every automorphism of $$A$$ sends $$B$$ to itself) and $$C$$ is a subalgebra of $$B$$ that is characteristic in $$B$$. Then, $$C$$ is a characteristic subalgebra of $$A$$.

Particular cases

 * Characteristicity is transitive
 * Characteristicity is transitive for Lie rings

Other similar facts

 * Full invariance is transitive for any variety of algebras
 * Derivation-invariance is transitive for any subvariety of the variety of rings
 * Derivation-invariance is transitive for Lie rings