Brought over from another thread, I'm of the opinion that influence doesn't need to be a separate stat (whether it be a single stat for the whole group or a stat each) and should be folded into the social system.
Important note straight away: I'm not opposed to the influence system - just the stat itself.
My main rationale is that at the moment, especially in the Beta, that influence and social stuff are working completely different systems. For example, someone could have the peer talent and be best buds with the mechanicus, but when it comes to requisitioning a bionic arm he's no better off.
My other rationale is that I see the social system as doing much the same thing as influence does; it's what you use to get your own way - and getting equipment and and assistance is a subset of that.
The influence system does reflect doing favours for people etc, so to cover I'd propose letting the players buy (or just give them) the peer talent for an organisation if they've done them a good turn.
I should also note that I envisage this as working in the context of a more expanded peer system - I dislike the idea that a generalised social character is much better at interacting with an organisation than a member of that organisation. So I'd also probably dish out peer as part of the backgrounds, and introduce further grades of peer for bigger bonuses.
I will finally note that I realise this is all just an extremely draft idea with a lot of holes in it - but I think it has the advantage of simplifying the system by removing the stat.