Summary: We compute the standard invariant of the `subgroup-subfactor' \(P\times_{\alpha|_H} H\subset P\times_\alpha G\), where \(\alpha\) denotes an outer action of a finite group \(G\) on \(\text{II}_1\) factor \(P\), and \(P\times_{\alpha|_H}H\) denotes the obvious crossed-product obtained by restricting the action to \(H\). We then use this description to exhibit a pair of non-isomorphic subgroups \(H_i\), \(i= 1,2\), of the symmetric group \(S_4\) such that the subfactors \(P\times_{\alpha|_{H_i}}\subset P\times_\alpha G\), \(i= 1,2\), are conjugate, thereby disproving a conjecture of \textit{K. Thomsen} [Factors and subfactors from ergodic theory, talk at Madras Conference (1997)] that `the subgroup-subfactor remembers the subgroup' (provided the subgroup contains no non-trivial normal subgroup of the ambient group).