Why did God command Abraham to be circumcised as well as all males from then on?

God told Abraham:
Genesis 17:11 You are to undergo circumcision, and it will be the sign of the covenant between me and you. (NIV)


So, it was for symbolic purposes as a sign of the covenant between God and Abraham.

Why do some groups of people add to this command and circumcise females as well?

I've never heard of any Christian groups doing this. It has been practiced in many parts of Africa as a rite of passage and some Muslims do support it although it is not found in the Koran, but I know of no group claiming to be Christian that has ever attempted to justify it on religious grounds.