Whether two groups in the form of infinite product are isomorphic

Question

Denote by $Z_n$ the cyclic group of order $n$. I want to determine whether two groups $G = Z_4 \times \prod_{i=1}^\infty (Z_2 \times Z_2 \times Z_4)$ and $H = (Z_2 \times Z_2) \times \prod_{i=1}^\infty (Z_2 \times Z_2 \times Z_4)$ are isomorphic or not. When observing the first factor outside of $\prod$, they seem not isomorphic. However, if I think about the map $$(x_0, (x_1^1, x_2^1, x_3^1), (x_1^2, x_2^2, x_3^2), (x_1^3, x_2^3, x_3^3), \cdots) \mapsto ((x_1^1, x_2^1), (x_1^2, x_2^2, x_0),(x_1^3, x_2^3, x_3^1), \cdots ),$$ then it looks like this map defined an isomorphism between two groups (an idea of thinking about changing the order of factors). Then why does such a problem arises at my first glance? Any comments are appreciated.