Algebras with superautomorphism: simple algebras and codimension growth

Let $A$ be an associative algebra endowed with a superautomorphism $\varphi.$ In this paper we completely classify the finite dimensional simple algebras with superautomorphism of order $\leq 2.$ Moreover, after generalizing the Wedderburn-Malcev Theorem in this setting, we prove that the sequence of $\varphi$-codimensions of $A$ is polynomially bounded if and only if the variety generated by $A$ does not contain the group algebra of $\mathbb{Z}_2$ and the algebra of $2\times 2$ upper triangular matrices with suitable superautomorphisms.