Two permutations are conjugated in $S_n$ iff they have same spectrum (as matrices). Since $A,B_1,B_2$ are symmetric, they are a product of $k$ (the same $k$ for all) disjoint transpositions.
Let $A,B_1,B_2$ be such matrices. Then there exist $X,Y\in S_n$ s.t. $X^TAX=B_1,Y^TAY=B_2$.
Does there exist $Z\in S_n$ s.t. $Z^TAZ=B_1,ZAZ^T=B_2$ ?
$\bullet$ I know a random example -when $n=5$- without solutions in $Z$.
$\bullet$ Conversely, if at least one such a $Z$ exists, I know $2$ random examples -when $n=8$- with $12$ or $16$ solutions in $Z$.