Example

If E is a G-set, each orbit of E, with the induced operation, is a homogeneous set under G.

Let G be a group and H a subgroup of G. Consider the set $G/H$ of left cosets modulo H. The group G operates on $G/H$ on the left by $(g, xH) \mapsto gxH$. Let N be the normalizer of H. The group N operates on $G/H$ on the right by $(xH, n) \mapsto xHn = xnH$. This operation induces on H the trivial operation and hence, on passing to the quotient, $N/H$ operates on $G/H$ on the right. Let $\phi : (N/H)^0 \to S_{G/H}$ be the homomorphism corresponding to this operation.