Subset sum problem for permutations
The subset sum problem is even harder(NP-hard) for special groups which you can embed into $S_n$. See the paper you have linked in the problem description.

$\textbf{Observation:}$ $g$ can be writtens as combination of elements $\{g_1,\ldots,g_k\}$ if and only if $g \in \langle g_1,\ldots,g_k\rangle$. Under the assumption that each of $g_i$ can appears any number of times.

The complexity of the problem depends upon the input representation. There are two two most commonly used ways, Cayley table and generating set. For Cayley table see this paper for results. Read the CGM(Cayley membership problem) Link

$\textbf{CGM} $

$\textbf{Input :}$ A group $G$ by its Cayley table, $X \subseteq G$ and $t \in G$.

$\textbf{Output :}$ Does $t$ belong to the subgroup $\langle X \rangle$ generated by $X$?

In general problem is in Symmetric log space.



$\langle A \rangle$ means subgroup generated by $A$