An (n, k)-perfect sequence covering array with multiplicity λ, denoted PSCA(n, k, λ), is a multiset whose elements are permutations of the sequence (1, 2, . . . , n) and which collectively contain each ordered length k subsequence exactly λ times. The primary objective is to determine for each pair (n, k) the smallest value of λ, denoted g(n, k), for which a PSCA(n, k, λ) exists; and more generally, the complete set of values λ for which a PSCA(n, k, λ) 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 Sn. This allows us to determine the new results that g(7, 4) = 2 and g(7, 5) ∈ {2, 3, 4} and g(8, 3) ∈ {2, 3} and g(9, 3) ∈ {2, 3, 4}. We also show that, for each (n, k) ∈ {(5, 3), (6, 3), (7, 3), (7, 4)}, there exists a PSCA(n, k, λ) if and only if λ ≥ 2; and that there exists a PSCA(8, 3, λ) if and only if λ ≥ g(8, 3). n k |P |, from which we obtain the necessary condition |P | = k!λ.