  Given a nilpotent Lie group $N$, a compact subgroup $K$ of automorphisms of
$N$ and an irreducible unitary representation $(\tau,W_\tau)$ of $K$, we study
conditions on $\tau$ for the commutativity of the algebra of
$\mathrm{End}(W_\tau)$-valued integrable functions on $N$, with an additional
property that generalizes the notion of $K$-invariance. A necessary condition,
proved by F. Ricci and A. Samanta, is that $(K\ltimes N,K)$ must be a Gelfand
pair. In this article we determine all the commutative algebras from a
particular class of Gelfand pairs constructed by J. Lauret.
