okay, my title may not be that good...

what I am thinking about is why would Nathan give Claire a second chance. and why would HRG be so surprised that Claire was on the plane. what I am thinking is that HRG made a deal, as he has done before, that he would help Nathan round up those with abilities in exchange for keeping Claire safe.

so now that Claire is on the plane, will HRG think that Nathan double crossed him? I mean lets be honest Nathan will go after anyone (even his brother). so will that lead to HRG helping the Heroes?

your basically right i believe
because Claire was not suppose to be on that plane
thats why he was surprised to see her on there

Only if Claire can think quickly and lie convincingly as to how she "escaped" being drugged and restrained like the rest of the specials on board.

Also, you're assuming that Claire would even want her father's help in this situation. Given the circumstances, I don't think she will want to even talk to Noah, let alone ask for his help. I'm expecting a nice dramatic line from Claire at some point during the episode like, "I used to have two fathers, now I have none."

It's possible Noah still might think Nathan betrayed him in any event because of how she is dressed.