Let $H,K \le A_n$ are two maximal subgroups of $A_n$. Let $A_n$ acts on the cosets of $H$ in $A_n$ by a map $\pi_1$ and $A_n$ acts on the cosets of $K$ in $A_n$ by a map $\pi_2$.
Question : How block systems corresponding to maximal subgroups are related?
As we know that set of cosets forms a block system i.e. ($S^\sigma = S$, $\forall \sigma \in A_n$) let S be set of all cosets of $H$ in $A_n$ and $Q$ be set oF all cosets of $K$ in $A_n$. Is it true that there exists a $\sigma \in A_n$ such that $\sigma^{-1}S\sigma = Q$? Or there exists an isomorphism form set $S$ to $Q$?
$G/H$ and $G/K$ are isomorphic as $G$-sets if and only if $H$ and $K$ are conjugate subgroups of $G$. The reason for the forward implication is that the stabilizers of the points in $G/H$ are exactly the conjugates of $H$. (In more detail, the stabilizer of the point $gH$ in $G/H$ is $gH(g^{-1})$.)
Conversely, if $H$ and $K$ are conjugate in $G$, say $c\in G$ with $(c^{-1})Hc=K$, then $G/H$ is isomorphic to $G/K$ (as $G$-sets) via the isomorphism $gH\mapsto gcK$.
All of this works for any group $G$ and any subgroups $H$ and $K$. There's noting special about $A_n$ or about maximal subgroups.