Induced representations of Lie groups in case of compact quotient

Question

Suppose you are given a split exact sequence of Lie groups $$ 1\to N\to G\to K\to 1, $$ where $K$ is compact abelian. If $(\sigma,V_\sigma)$ is an irreducible unitary representation of $N$, is it true that the induced representation $Ind(\sigma)$ is a direct sum of irreducibles, each occuring with finite multiplicity?

Here the induced representation is the right-translation representation on the space of all measurable functions $f:G\to V_\sigma$ with $f(nx)=\sigma(n)f(x)$ and $\int_K ||f(k)||^2\,dk<\infty$ modulo nullfunctions.