Kin selection is not a special case of evolution, it is the main case. Genes are types not tokens, and if you can benefit more than two brothers at the expense of yourself, it is the adaptive thing to do.
However, kin selection drops off exponentially: two brothers, four grandparents, eight cousins...
So kin selection does not explain altruism in societies. It is not adaptive to die in war for your society, only for your brothers (grandparents, cousins...). Group selection does not occur, because a selfish person can benefit by not going to war while his neighbours so.
But group selection appears to occur. Societies must cooperate to win wars, and the losing, less cooperative society is often eliminated entirely. (Melanesian men conquered Vanuatu and Tonga, killed all the previous male inhabitants and took their women.)
So what is going on? The answer is that the only feature which is selected by group selection is cooperation. Cooperation is not altruism, it is done selfishly to benefit you and your kin. That it benefits the other party is incidental. Ability to cooperate is a property of individuals, not groups, though it only makes sense in a group setting (you need other people to cooperate with).
You die in war to benefit your kin. You devise social enforcement mechanisms to ensure your neighbours contribute to the war effort.
You need a minimum number of cooperators in your society to win wars. Societies which are better at cooperation will outcompete societies which are not. Asabiyyah matters. In-fighting at the expense of out-fighting will bring down a society. Clearly, if a cooperative society exterminates a non-cooperative society and takes its land, genes for cooperation will spread.
Group selection cannot occur if it conflicts with individual/kin selection. Group selection does not occur often enough to build features such as cooperation, it only increases their number/propagates them.
Why is cooperation the only feature selected by group selection? Because the group is the unit of cooperation. That's what a group is.