Is a wife's separation from her husband, and taking her children to her family's home, a sound action after suspecting his betrayal?

The default is to assume the best of a Muslim and interpret their actions in the most favorable light. This is particularly emphasized between spouses. If what you have mentioned does not go beyond the stage of doubt and suspicion, then you must cast all of that aside. However, if it is proven that there is a relationship between your husband and another woman, we believe that you should continue trying to reform him and rescue him from what he has fallen into, out of kindness to him and to preserve the family unit, especially since there are children. This can be done by involving two righteous men from your family and his. If your husband's condition improves, then the objective has been achieved. Your husband should know that there is no way for you to remain with him except by cutting off his illicit relationship with this woman. If he persists in his previous state, then ask him for a divorce.