Can you expiate someone else's wrongs?
Solution 1:
Of course you can. The most notable usage of the word, in fact, is that exact scenario: Jesus Christ's suffering for the sins of the world in order to provide redemption. "Expiate" is a Latin derivative, and where we often say "atone" in English, romance languages use the word "expiate" instead. The Expiation, "expiation" with a capital E, is what Catholics call Jesus Christ's suffering for the sins of the world.
https://www.thousandtongues.org/blog/general/the-expiation-of-jesus-christ
https://www.gotquestions.org/expiation.html
A non-religious example of expiation would be me paying your drunk-driving fines. I am paying the reparations of your crime. In the United States, though, it's actually illegal to do that, but I digress.
A second non-religious example would be parents paying for a window their child broke while someone else was charged with their child's care. In so doing, they are expiating their child's wrongdoing, or if not their child's, the caregiver's.