HINT: If $A$ and $B$ have the same cardinality, and $f\colon A\to B$ is a bijection, then whenever $R$ is a relation on $B$ there is some $R'$ which is a relation on $A$ such that $f$ is an isomorphism between $(A,R')$ and $(B,R)$.
This $R'$ is defined in the most obvious way. If $R$ is a set of $k$-tuples then we define: $$R'=\{\langle a_1,\ldots,a_k\rangle\mid\langle f(a_1),\ldots,f(a_k)\rangle\in R\}.$$
Now recall that an binary operation is merely a binary function, or a trenary relation. Find some countably infinite group $(G,\cdot)$ and define $*$ accordingly.