Groups as categories

Recall that to any group we can associate a category $\mathbf{B}G$ that has a unique object, an arrow for each element of $G$, and arrow composition given by the group law in $G$. (See a bit more here.)

1. Show there is a correspondence^[1] between groups and one-object categories in which every arrow is invertible.
2. Suppose $G$ and $H$ are groups. Show that group morphisms $\varphi :G\to H$ correspond^[2] to functors $\mathrm{\Phi }:\mathbf{B}G\to \mathbf{B}H$.
3. Recall that a permutation representation of $G$ is a group morphism $\varphi :G\to S_X$, where $S_X$ is the permutation group on a set $X$. Show that permutation representations of $G$ correspond to functors $\mathbf{B}G\to \mathbf{S}\mathbf{e}\mathbf{t}$.
4. Suppose $\varphi ,\psi :G\to H$ are group morphisms, with corresponding functors $\mathrm{\Phi },\mathrm{\Psi }:\mathbf{B}G\to \mathbf{B}H$. Show there is a natural transformation $\mathrm{\Phi }\Rightarrow \mathrm{\Psi }$ if and only if $\varphi$ and $\psi$ are conjugate, i.e., there is an element $h\in H$ such that $\psi (g)=h(\varphi (g))h^{-1}$ for all $g\in G$.