Let $K\subseteq G$ be compact Lie groups, with $G$ connected. Suppose that $H\_1,H\_2\subseteq K$ are two closed subgroups of $K$ that are conjugate in $G$. Are they conjugate in $K$?



Conjugacy in compact Lie groups
In general the answer is no. For example, in $U(n)$ you have the subgroup $U(1)^n$ which has many copies of $U(1)$ in (by choosing various "coordinate lines" and act trivially on the orthogonal complement). These specific copies of $U(1)$ are all conjugate in $U(n)$ by permutation matrices, but they are not conjugate in $U(1)^n$ since it is commutative. 


