SUPPORT

I think for a question like this you're best off going to the administrators of the group and asking what their policies/rules are in such situations.

Did he enter before or after he was kicked?

don't send it have them reroll it.

If said winner joined giveaway before he was kicked, and the giveaway not stated any rules, you are obligated to send the game to him.
If said winner joined giveaway before he was kicked, and the giveaway stated rules, you can refuse to give depending on the rules
If said winner joined giveaway after he was kicked, you can refuse to give.

If the situation was the first and you refuse to give, the group will appear to be extremely shady and dishonest

It's the gifter's call. The gifter holds the gift and can choose to either ask for a reroll or simply gift the game. Would any of you really force a gifter(somehow) to give the gift to someone else? It's a rare situation for a gifter to deal with such things but it all boils down to the gifter's decision and no one else. Were it a public giveaway then obviously that's another story.

He won the gift, send it to him. Else I'd be happy if the admins ban you for not honoring your giveaway.

Can you prove that somebody from that group who also wanted the game didn't ask the group admin to kick the winner out ? No, you can't.

SEND THE GAME !

People keep saying that you are obligated to send the gift. No, gift giving is voluntary, however if you don't give a gift you get negative feedback, and if there was never any intention of giving a gift in the first place (fake giveaway) you can be banned.
As for private and group (in particular private groups) giveaways, the mods I am fairly sure are likely to do a re-roll if asked for them if the winner doesn't meet a certain criteria eg. Winning a giveaway for a group they were removed from.

I think the best way to sort this out is gifting me the game.

Because saying 'thanks' every giveaway is really necessary(talking about his 'three comments')