Consider groups are finite.
Let $G_1 = A \rtimes_{\phi_1} B_1$ and $G_2 = A_2 \rtimes_{\phi_2} B_2$. Note that $A_1,A_2,B_1,B_2$ are cyclic groups. It is also known that $A_1 \cong A_2$ and $B_1 \cong B_2$. $\rtimes$ denote the semi-direct product
I need to check whether $G_1 \cong G_2$ or not. I tried to think, it seems to me that it will depends upon the $\phi_1$ and $\phi_2$. In the case direct product we have remak-krull schmidt but for semi-direct product, I don't have any tool.
Question : When $G_1 \cong G_2$ for the above question?
Edit : My Question is what is the condition such that $G_1 \cong G_2$ which is necessary and sufficient also.
No, $G_1$ need not be isomorphic to $G_2$. Take the two groups of order $12$, $G_1=C_3\rtimes_{\phi_1}C_4=C_3\times C_4$, with trivial $\phi_1$ and $G_2=C_3\rtimes_{\phi_2}C_4$, with non-trivial $\phi_2$. Then $G_1$ and $G_2$ are not isomorphic, because $G_1$ is abelian, but $G_2$ is not. If $\phi_1$ and $\phi_2$ are conjugated, then $G_1\cong G_2$. See also the following question:
$ K \rtimes_{\phi_1} C \cong K \rtimes_{\phi_2} C$ when $\phi_1(C), \phi_2(C)$ are conjugated and $C$ is a product of two cyclic groups