If I have already made havdala at the beth ha-kneset, and yatzati yedei chova, it is a “problem” if I make havdalah again in my house for my wife? Why we can’t say that even if I have fulfilled my obligation, kol israel arevim ze la ze, and I can make havdalah again for her?
If you have already made havdalah in shul, it is better not to make havdalah for your wife, because of the opinion that women are not obligated in havdalah. Therefore, it is better that she should make havdalah for herself. However, if this is difficult, you can make havdalah for her, because the principle opinion is that women are obligated in havdalah.
It is therefore better to have intention not to be yotzei one’s obligation of havdalah in shul, and to make havdalah at home.
Sources: See Shulchan Aruch 296:8; Mishnah Berurah 296:35-36; Beis Yosef (citing from Rambam and from Orchos Chaim).