On the contrary, God sets Himself up as sole avenger and forbids it to His servants. Read Deuteronomy 32:35, Romans 12:19 and Hebrews 10:30.
You are correct except in the avenging...God Judges and until then by and through His covenant brought to us by His son Jesus He forgives us when we repent. If we do not repent, He does not take revenge upon us..He allows us His complete absence...in hell, but that is our choice...not His avenging.