Well........considering that Abraham's wife Sarah had a servant from Egypt who was Hagar that bore Abraham his first son (who was Ishmael). When there was a bit of strife in the tribe (Hagar was trying to take over Sarah's position), Sarah told Abraham that Ishamael and Hagar had to leave, and they were kicked out of the tribe.
Interestingly enough, Hagar was an Egyptian princess. When they were kicked out, according to Islam, Hagar went over to a bush to pray, out of sight of Ishmael.
Why?
I personally think that because Hagar had gotten kicked out of Abraham's tribe, she wasn't necessarily praying to the God of Abraham but rather some Egyptian diety, otherwise, why would she hide herself from her son when she was praying?
Probably because she knew that Ishmael would stop her, as Ishmael would have known the difference just as Abraham did, and Hagar was probably pissed at getting kicked out, so I don't think she would use anything from Abraham's religion.
I do not believe that Allah is the same as HaShem either, and I'm a Taoist that studies Judaic and Christian theology, as well as have investigated a bit of Islam.