Summary execution, early on in an apocalyptic scenario, I find hard to believe, and I'm speaking as a veteran (active-duty MP) and an ex-cop.
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I just have a huge believability problem with summary executions that early. They'll detail soldiers out to stand guard over a barbed wire enclosure before they start shooting people. Every soldier in the Army has served guard duty for one reason or another. My son's a mechanic with 1st Infantry in Kansas, and has spent more time on the front gate than in the motor pool in the last six months.
I agree with this. I served five years active duty with the U.S. Army (Infantry), followed by seven with the Texas Army National Guard (also Infantry). I spent two of those seven years activated or deployed--six months with the airport security mission after the 9-11 attack, and eighteen training up and deploying to Iraq in 2004-05.
If the National Guard in your story is anything like the Guard I served with, summary execution is very unlikely. If the unit has a strong backbone of combat veterans and has to attack the rebels in order to restore order, they are
very likely to kill rebels in the assault. If they capture the rebels alive, they are very unlikely to execute them, especially without trial.
Some factors that might affect this likelihood:
1. The Guard unit has been under fire a lot lately, especially if they believe these are part of the group that's been shooting at them. Your description doesn't make it seem like that's the case.
2. The situation is uncertain locally as well as nationally. If the unit believes there is no law anymore except the law it brings, it might be more likely to take drastic measures. This becomes more likely if combined with #1.
3. The Guard platoon is part of a much larger unit (Brigade at least and well-equipped), whose commander has decided martial law means
he is the law and given orders to that affect, and whose soldiers are unquestioningly loyal to him. This is unlikely unless 1 and 2 are also met, and this early in the scenario.
It's also important to remember that National Guard soldiers largely come from among the populations they will be mobilized to protect. Their families will be among the refugees. That means brutal tactics are unlikely in general, but if the Guard unit believes the rebels threaten their families, the rebels are likely to have a hard day of it.
Of course, that also makes other problems like loss of discipline, and even desertion, more likely as soldiers decide they need to be home protecting their families. The longer the crisis persists, the greater a problem this will become.
You can believably go either way if you set it up properly, but this early after the disaster it would be much less likely than later.
Good luck!
HN