A v4 pool may name a hook: a contract called at fixed moments of a swap, a deposit or a withdrawal.
Which moments is written into the hook's address, and the protocol enforces it.
A hook allowed to act on swaps can change what they cost, so figures read from the curve alone may not hold for that pool.
A hook's permissions are fixed when the pool is created; the pool cannot change which hook it names.
What a hook does with a permission is in its code, which this site does not read and does not vouch for.
The hook directory here lists the hooks behind the week's busiest v4 pools, with what each is permitted to do.