functoriality of the Burnside ring

We wish to show how the Burnside ring $\mathrm{\Omega }$ can be turned into a contravariant functor from the category of finite groups into the category of commutative, unital rings.

Let $G$ and $H$ be finite groups. We already know how $\mathrm{\Omega }$ acts on objects of the category of finite groups. Assume that $f:G\to H$ is a group homomorphism. Furthermore let $X$ be a $H$-set. Then $X$ can be naturally equiped with a $G$-set structure via function:

$$(g,x)⟼f(g)x.$$

The set $X$ equiped with this group action will be denoted by $X_f$.

Therefore a group homomorphism $f:G\to H$ induces a ring homomorphism

$$\mathrm{\Omega }(f):\mathrm{\Omega }(H)\to \mathrm{\Omega }(G)$$

such that

$$\mathrm{\Omega }(f)([X]-[Y])=[X_f]-[Y_f].$$

One can easily check that this turns $\mathrm{\Omega }$ into a contravariant functor.