Why does the sniper kill the old woman?

The sniper is fighting on one side (Republican) of the Irish Civil War. From a rooftop, he foolishly lights up a cigarette. The smoke signals his position and the enemy sniper fires at him. On the street below, an armored car approaches. The old woman signals to a soldier from the armored car and points out the position of the sniper. She is an "informer." The sniper is trying to avoid being killed. It is a war zone. So, he fires and kills the soldier looking out of the turret. Then he kills the woman, even though she is running away. The sniper reasons that the woman would simply tell another enemy soldier of his position. He kills her in order to keep the enemy from knowing exactly where he is. 

The other, more fundamental, reason the sniper kills her is that he has become accustomed to killing. The narrator describes him as having the face of a student, but one who has become used to violence, war, and death. In this, his adopted mentality of the soldier, he does not hesitate in killing her. His automatic thinking is "kill or be killed."