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Is there equivalent of this Hindi expression
Aag (Fire) bhi (also) lagana (to set/ignite) , Bhujane (Extinguish) bhi (also), Khudhi (self) Jana (to go)
which means
To set a fire and also to go and extinguish it
Which means a person sets the fire intentionally and in pretense goes back to extinguish it too (as in save the day) to appear like a good Samaritan.
That is what false flag means, except that is usually when a state or authority is involved, and an excuse for military action. We sometimes talk about a bogeyman as something created in the same sense, to make its creator appear as protector. We say someone is tilting at windmills when imagining an enemy, straight from Don Quixote.
As a mental health issue, there is "Munchausen syndrome by proxy," in which a caregiver intentionally harms their ward---so that they can then help them to recover. This lets the caregiver feel wanted (and useful).
(In criminal terms, what you describe could be seen as a type of "forensic countermeasure," where the perpetrator tries to shift suspicion from themself by doing something that they think a criminal wouldn't normally do: return to the scene and provide aid.)
At least in popular media, fictional characters who commit crimes---on the pretence of helping---have been dubbed to have a (coincidentally with what you said) "good Samaritan" motive. But that can either be about ending somebody's suffering or about inflating the criminal's own need for attention and validation.
Perhaps "hero complex," used in a criminal sense, would also apply.
A more accurate response would depend on if the Hindi expression relates to the act itself or a motive behind it.
I'm having a really hard time thinking of an idiom that people use to describe this situation. The translation you gave for your Hindi idiom would definitely be immediately intelligible to English speakers if you refined it a bit. As an American, it's not uncommon to hear my coworkers saying they are "putting out fires" when they are dealing with server problems or issues in the office. I could see myself using it in a sentence like:
There he goes, putting out fires he started himself.
or
My manager is notorious for starting fires just so he can put them out himself.
a base motive TFD and here psychology wikia
Motivation for some action or actions that is guided by selfishness or moral depravity.
hero syndrome wikipedia and hurt and rescue changing minds may be helpful
'A drowning person will clutch at a straw', so push them in the water, then throw them a rope.
