Anattakatāni kammāni kathamattānaṃ phusissantī'ti
"Then what 'self' can those actions affect, which are not self-done?"

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This is the reflexion of a Bikkhu to Buddha's lesson on the five aggregates subject to clinging (appropriation) - when Buddha states that these aggregates are not "ourselves". (SN 22.82 + SN 22.33 - SN 22.59)
Therefore the Bhikkhu wonders what "self" is responsible of kamma (actions/deeds), if the self (atta/satta) is not ourselves?
Buddha restates the impermanence of the khandhas; without mentioning that "self". Why?

Simply because that true "self" (or Self), if it exists, can only be conceived through the riddance of kamma - of any saṅkhāras (determinations/synergies/volitional formations) - of any khandhas.
In other words, as long as kamma (action) is involved, it is the illusory (material/worldly) self - viz. atta/satta - that is responsible. Responsible of its kamma (action); as well as accountable of its kamma (reaction,) and saṃsāra.

But why (and how,) is satta responsible?
Because nāmarūpa needs some feedback on its "creations". And for that matter, this also necessitates that nāmarūpa entitles satta (through saḷāyatana - the spheres of senses,) with a certain degree of freedom. Otherwise, nāmarūpa would answer itself; and the need for saḷāyatana would be useless.
Satta is here "to be felt" (SN 12.37), with a genuine feedback - might it be delusory.

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