  The study of $G$-equivariant operators is of great interest to explain and
understand the architecture of neural networks. In this paper we show that each
linear $G$-equivariant operator can be produced by a suitable permutant
measure, provided that the group $G$ transitively acts on a finite signal
domain $X$. This result makes available a new method to build linear
$G$-equivariant operators in the finite setting.
