I am fairly new to group operations and given with this problem. I did not manage to come up with anything meaningful - Hence, I am kindly asking for support. Thanks!
Let $(G, \cdot)$ and $(H, +)$ be groups, and $G$ be isomorphic to $H$.
Show that $(G \times F^*_3, \circ)$ and $(H \times \text{sym}(2), \oplus)$ are isomorphic,
where $\circ : (G \times F^*_3) \times (G \times F^*_3)$ , $(a_1, a_2) \circ (b_1, b_2) := (a_1 \cdot b_1, a_2 \cdot b_2)$,
represents the component-wise group operation on $G$ and $F^*_3$,
and $\oplus : (H \times \text{sym}(2)) \times (H \times \text{sym}(2))$ , $(a_1, a_2) \oplus (b_1, b_2) := (a_1 + b_1, a_2 \circ b_2)$,
represents the component-wise group operation on $H$ and $\text{sym}(2)$