  An $(n,k)$-perfect sequence covering array with multiplicity $\lambda$,
denoted PSCA$(n,k,\lambda)$, is a multiset whose elements are permutations of
the sequence $(1,2, \dots, n)$ and which collectively contain each ordered
length $k$ subsequence exactly $\lambda$ times. The primary objective is to
determine for each pair $(n,k)$ the smallest value of $\lambda$, denoted
$g(n,k)$, for which a PSCA$(n,k,\lambda)$ exists; and more generally, the
complete set of values $\lambda$ for which a PSCA$(n,k,\lambda)$ exists. Yuster
recently determined the first known value of $g(n,k)$ greater than 1, namely
$g(5,3)=2$, and suggested that finding other such values would be challenging.
We show that $g(6,3)=g(7,3)=2$, using a recursive search method inspired by an
old algorithm due to Mathon. We then impose a group-based structure on a
perfect sequence covering array by restricting it to be a union of distinct
cosets of a prescribed nontrivial subgroup of the symmetric group $S_n$. This
allows us to determine the new results that $g(7,4)=2$ and $g(7,5) \in
\{2,3,4\}$ and $g(8,3) \in \{2,3\}$ and $g(9,3) \in \{2,3,4\}$. We also show
that, for each $(n,k) \in \{ (5,3), (6,3), (7,3), (7,4) \}$, there exists a
PSCA$(n,k,\lambda)$ if and only if $\lambda \ge 2$; and that there exists a
PSCA$(8,3,\lambda)$ if and only if $\lambda \ge g(8,3)$.
