A moment of inertia tensor $T$ can be rotated by a rotation matrix $R$ to $T'$ using $T' = RTR^{-1}$. If I know $T$ and $T'$, how can I find the rotation $R$ such that I can apply that rotation to another tensor $A$ to take it to $A' = RAR^{-1}$?
Basically, I am trying to use the moment of inertia tensor to find the rotation that has been applied to an object $T$ to apply the same rotation to another tensor quantity (in this case the Born Effective Charge) $A$ so that it is oriented correctly.
Given two MMOI tensor values measured at two different orientations (basis vectors), and assuming both are summed about the same point. Lets call the two tensors ${\bf T}_A$ and ${\bf T}_B$ for the two frames A and B and note that there exists a unique frame C such that ${\bf T}_C = {\rm diag}(I_1,I_2,I_3)$ designating the principal MMOI values.
I steer away from using apostrophes for designating frames as it confuses the notation.
Take each MMOI tensor and perform an eigenvalue decomposition to find the directions of the principal axis
$$ {\bf T}_A = {\rm R}_{C/A} {\bf T}_C {\rm R}_{C/A}^{-1} $$
where ${\rm R}_{C/A}$ designates the rotation of frame C with respect to A.
To make sure ${\rm R}_\ldots$ are rotations you have to enforce ${\rm R}^{-1} = {\rm R}^\intercal$ in the eigenvector decomposition process.
Similarly
$$ {\bf T}_B = {\rm R}_{C/B} {\bf T}_C {\rm R}_{C/B}^{-1} $$
where ${\rm R}_{C/B}$ designates the rotation of frame C with respect to B.
Now you are asked to find ${\rm R}_{A/B}$ such that ${\bf T}_B = {\rm R}_{A/B} {\bf T}_A {\rm R}_{A/B}^{-1}$ which is simply
$$ {\rm R}_{A/B} = {\rm R}_{C/A}^{-1} {\rm R}_{C/B}$$