Polynomial identities, central polynomials and cocharacters of $M_2(F)$ with $G$-graded involution

Let $F$ be a field of characteristic zero, $G$ be a finite abelian group and $M_2(F)$ be the algebra of $2\times 2$ matrices over $F$. In this paper, we consider all $G$-graded involutions on $M_2(F)$ and determine the generators of the $T_{(G,*)}$-ideal of identities and of the $T_{(G,*)}$-space of central polynomials of $M_2(F)$. Moreover, we explicitly compute the $\langle n\rangle$-cocharacter and the $n$-th $(G, *)$-codimensions for most of the gradings. For the case of gradings by the Klein group, the $(G, *)$-central codimensions and proper central $(G, *)$-codimensions are also explicitly computed.