This is the 14378th most frequent Russian word.
отомстить
"To take revenge."
Here, 'отомщу' is used to express a planned action of vengeance or retaliation for the friends, signifying direct personal involvement in the act of avenging.
Я отомщу за своих друзей.
I will avenge my friends.
In this sentence, 'отомстил' indicates a completed action in the past, where the subject carried out an act of revenge or retribution for the mentioned insults.
Он отомстил за нанесённые оскорбления.
He avenged the inflicted insults.
Here, 'отомстить' is in its infinitive form, used hypothetically about potential future actions by another entity to seek revenge.