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In this note we make some remarks concerning maximum principles holding for nonlocal diffusion operator of the form $$\mathcal{M}[u](x):={\int }_{G}J(g)u(x\ast g^{-1})d\mu (g)-u(x),$$ where $G$ is a group acting continuously on a Hausdorff space $X$ and $u\in C(X)$.

We first investigate the existence of a strong maximum principle in the general situation and then focus on the case of homogeneous spaces. Depending on the topology of the homogenerous space, we give contidions on $J$ and $d\mu$ such that $\mathcal{M}$ achieves a strong maximum principle. We also revisit the classical case of convolution operator on ${\mathbb{R}}^n$.