Is it permissible for the husband of the questioner's sister to take her back after divorcing her by phone while he is traveling abroad, and what are the necessary conditions for that?

Divorce by phone is valid. Your brother-in-law can take her back before her waiting period (iddah) ends, which is: her purity from the third menstrual cycle, or three months if she does not menstruate, or delivery of her baby if she is pregnant. This is permissible if this is the first or second divorce. The reconciliation (raj'ah) is effected by saying: "I take you back" or "I return you to my marital bond." However, if this was the third divorce, she is unlawful for him and cannot become lawful for him again unless she marries another man willingly and validly, and then he divorces her after consummating the marriage.