Size of two conjugate subgroups of a finite group Given a finite group $G$ and any subgroup $H$. In all the examples I've looked at the intersection of two different conjugate subgroups of $H$ always had the same size. Is this  always the case? I know that the intersection $H^{g_1} \cap H^{g_2}$ has the same size as $H \cap H^{g_2g_1^{-1}}$, but that's a bit of how far I got.
 A: $\newcommand{\Set}[1]{\left\{ #1 \right\}}$$\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$There might be a simpler example, still the following one is somewhat natural.
Consider the abelian group $A$ of order $2^{6}$,
$$
A = \Span{a_{1}} \times \Span{b_{1}} \times \Span{a_{2}} \times \Span{b_{2}}
$$
where $a_{1}, a_{2}$ have order $4$, and $b_{1}, b_{2}$ have order $2$. Note that $c_{1} = a_{1} b_{1}$ and $c_{2} = a_{2} b_{2}$ have order $4$.
Consider the automorphism $g$ of $A$ which maps
$$
a_{1} \mapsto c_{1},
c_{1} \mapsto a_{1},
a_{2} \mapsto c_{2},
c_{2} \mapsto a_{2}.
$$
An alternative description of $g$ is
$$
a_{1} \mapsto c_{1},
b_{1} \mapsto b_{1},
a_{2} \mapsto c_{2},
b_{2} \mapsto b_{2}.
$$
Consider the automorphism $h$ of $A$ which maps
$$
a_{1} \mapsto a_{2},
c_{1} \mapsto c_{2},
a_{2} \mapsto a_{1},
c_{2} \mapsto c_{1}.
$$
An alternative description of $h$ is
$$
a_{1} \mapsto a_{2},
b_{1} \mapsto b_{2},
a_{2} \mapsto a_{1},
b_{2} \mapsto b_{1}.
$$
Then in the natural semidirect product $A \rtimes \Span{g, h}$ the four subgroups
$$
\Span{a_{1}}, 
\Span{c_{1}}, 
\Span{a_{2}}, 
\Span{c_{1}}
$$
form a conjugacy class, but 
$$
\Span{a_{1}} \cap 
\Span{c_{1}} = \Span{a_{1}^{2}}
$$
has order two, while
$$
\Span{a_{1}} \cap 
\Span{a_{2}} = \Set{1}.
$$
