The conjecture that β(N,l) depends only on l

Let $N\geq 2$ and $1<l\mid N$. The homomorphism $\sigma_{N,l}$ has image $\operatorname{im}(\sigma_{N,l})=\beta(N,l)\mathbb{Z}$ for a unique positive integer $\beta(N,l)$. Dependence conjecture.

$$\beta(N,l)=\beta(l,l).$$

This would reduce the study of $\beta(N,l)$ to the diagonal values $\beta(l,l)$; the claim was verified for $l,N<1000$ by a SageMath program, but no proof or disproof is supplied here.