I don't have enough reputation to comment on this neat answer so I ask here for some hints/answers. I take the set $A = \{0,1,2,3\}$ and so according to this answer, $A \sim \mathbb{Z}_4$. My question is: does that certain bijection allow to write the subset $\{0,2\}$ of $A$ is isomorphic to the subgroup $\{0,2\}$ of $\mathbb{Z}_4$?
I asked this because I don't know the explicit rule of the bijection. Thanks.
If $A\sim\Bbb Z_4$ means that $A$ is isomorphic to $\Bbb Z_4$, then the statement doesn't make sense, since $A$ is simply a set, not a group. Given any bijection $b\colon A\longrightarrow\Bbb Z_4$, you can use it to define a group structure on $A$ by transport of structure: if $a_1,a_2\in A$, $a_1\oplus a_2=b^{-1}\bigl(b(a_1)+b(a_2)\bigr)$. Then $(A,\oplus)$ is a group which is isomorphic to $(\Bbb Z_4,+)$. And $\{0,2\}$ may be or may not be a subgroup of $(A,\oplus)$; it depends upon the choice of $b$.