$*$ is closed on $\{-2, 2\}$, and is well-defined.
Associative: yes
Identity: yes, we have $-2$ is an identity (with respect $*$): since $\max\{-2,g\}=g$ for any $g \in G$.
Closed under Inversion: NO: $\;2 \in G$ has no inverse, i.e. there is no such $g \in G$ that $\max\{g,2\}=-2$.
The failure of any one of the above conditions negates the prospect of $(G, *)$ being a group. Since closure under inversion fails, $(G, *)$ fails to be a group.
It's perhaps beside the point, but $*$ is commutative on $G$, and the group is finite, clearly (exactly 2 elements in $G$), so you actually have a finite, abelian monoid!