EDIT: I have realised I made a mistake when decompsoing the morphisms of $\mathscr B_G$.
Nevertheless, the question seems to be interesting on its own, so i will leave it. I would also like to cite this question from MO, which is a sort of converse to my question.
I shall consider a finite group $G$ and $\mathscr B_G$ will be the Burnside category with objects fintie $G$-sets and morphisms equivalence classes of spans $X\leftarrow W\to Y$ (and then taking group completion). Two spans $X\overset{f}{\leftarrow} W\overset{g}{\to} Y$ and $X\overset{f'}{\leftarrow} W\overset{g'}{\to} Y$ are equivalent if and only if there is an equivariant bijection $u:W\to W'$ such that $f=uf'$ and $g=ug'$. Finally, composition in $\mathscr B_G$ is done via pullback: $[X\leftarrow V\to Y]\circ[Y\leftarrow W\to Z]\overset{def}{=}[X\leftarrow V\times_Y W\to Z]$.
The neatest definition of a Mackey $\underline M$ is as an additive contravariant functor $\mathscr B_G \to \mathbf{Ab}$.
I am trying to break this definition down into pieces to recover the usual one with all the transfers and restrictions. Object-wise, it suffices to study Mackey functors on orbits $G/H$, $H\subset G$. I was wondering about the morphisms when the following question about group theory came up:
Question: Let $H\subset G$ be a subgroup, and consider $g_1,g_2\in G$ different elements. Is there exists a group $K$ such that the diagram below is a pullback diagram and $K\subset g_1Hg_1^{-1}$ and $K\subset g_2Hg_2^{-1}H$?
$\require{AMScd}$ \begin{CD} G/H @>{c_{g_1}}>> G/g_1Hg_1^{-1}\\ @V{c_{g_2}}VV @VVp_1V\\ G/g_2Hg_2^{-1} @>>p_2> G/K \end{CD}
[The maps $p_i$ are induced by the inclusions $K\subset G/g_iHg_i^{-1}$.]
More details:
A general morphism $\varphi:G/L_1\to G/L_2$ in $\mathscr B_g$ is defined by a sum in $H$ of spans of the form $G/L_1\overset{f_1}{\leftarrow} G/H \overset{f_2}{\rightarrow} G/L_2$. Now, each $f_i$ consists of a conjugation map $c_{g_i}$ followed by the canonical map $q_i:G/g_iHg_i^{-1}\to G/L_i$. If such a subgroup $K$ existed, then $\varphi$ would consist of sums and compositions (in $\mathscr B_G$) of ''elementary'' morphisms of the form
$G/K\overset{p}{\leftarrow} G/H \overset{q}{\to}G/L, \quad$ with $K,L\subset H$ and $p$ and $q$ induced by those inclusions.