Is it lawful in Islam to take revenge on those who spread unfounded rumours against Muslims?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Allah Almighty legalized taking vengeance from the wrongdoers by the appropriated criterions. He didn't allow it to every person to take revenge. Allah mentions a killer, and as we know the killing is the biggest unjustice which can be committed on the Earth after the disbelief: {… And whoever is killed wrongfully (Mazluman intentionally with hostility and oppression and not by mistake), We have given his heir the authority [to demand Qisas, - Law of Equality in punishment - or to forgive, or to take Diyah (blood-money)]. But let him not exceed limits in the matter of taking life (i.e. he should not kill except the killer). Verily, he is helped (by the Islamic law).} [17:33].
Allah didn't permit to victim's curator to take vengeance directly and immediately, but He ordered that suit against the murders should be brought to the authorities to give the guardian his right from the killer to take the "blood money" or, if he is pleased, to take revenge.
So if we see this procedure in the act of killing how can some individuals take the rights for the whole "Ummah" (Muslim nation) from the person who makes capital common offense to the Islamic Nation?
So if the Muslim knows that someone spreads false rumors against the Muslims, his/her duty is to institute legal action with the authorities against that person; they should take proper legal judgment for all that person's wrong deeds and words.
Allah knows best.