An adverb for performing a punitive action on someone who is actually not guilty?

Question

What's the right adverb for describing a punitive action that is done when the one being punished is actually not guilty?

For example,

Those who had been ___________ly thrown into prisons and withered in them for years, were all now suddenly released.

Father often gave his sons a good deal of spanking ______________ly, which took its own tally on him when they grew up.

All those cases, in which the judge had issued a capital punishment ____________ly, weighed on him now greatly.

(I place ly in the end of each blank because adverbs usually end that way in English; however, if there is an adverb of some other form, that will be fine, too)

Answer 1

"Wrongfully" would be the word I'd expect in the first sentence; possibly "erroneously", but I think wrongfully is more common for false imprisonment.

https://www.vocabulary.com/dictionary/wrongfully

Answer 2

unjustly would work:

characterized by injustice : unfair
// The convict received an unjust sentence.

(source: Merriam-Webster)

That doesn't really help you if you don't know what injustice means:

absence of justice : violation of right or of the rights of another : unfairness