We cannot discuss the specific case to which you refer, as we do not know its particulars. However, we must speak strongly about the phenomenon you describe.
One of the people whom we are most required to help, by searching for a legitimate halachic remedy and otherwise, is an aguna. An aguna[1] is a woman who is unable to live with her husband, either because he is missing or because they are incompatible, but to whom she is halachically “chained,” because she lacks a get or evidence of his death. Although her main, practical problem is that she is unable to remarry, the feeling of helplessness from which she suffers creates one of the most tragic situations that exist. Only someone who has been personally involved in such a situation can fully appreciate its severity.
At times, a woman can be an aguna without anyone being at fault (e.g., the husband is in an irreversible coma). However, some women are in this horrible predicament because their husbands are spiteful or have monetary or other demands. This behavior is unacceptable! It is as morally wrong as the behavior of one who stalks his ex-wife because of his animosity toward her! If a husband has grievances against his wife, he may raise them in court, preferably a beit din. The beit din may side with him; it may side against him. However, the Jewish community should not tolerate a husband taking the law into his own hands by withholding a get.
At different times and places in history, religious courts had the ability to physically coerce a stubborn husband to give a get, when a get was mandated in the most clear-cut manner. (There are husbands in Israel who are in jail for that purpose.) In cases that are a little less clear-cut, a harchaka d’Rabbeinu Tam can be employed. This is a painful form of publicly shunning the husband, not only in shul, but also in commercial and personal settings.[2] Withholding aliyot is only a simple means of pressure. Nowadays, outside of Israel, the main recourse is usually moral and (moderate) public pressure. Tragically, weak public response often results in the husband feeling only mild or, in some cases, no pressure.
Again, we cannot comment on the specific case that you raise. However, if the rabbinical courts have instructed your acquaintance to give a get and your rabbi has been asked to ensure some form of communal disapproval, then that is the very least that should be done. Others should follow the rabbi’s lead, not question it, and should not allow their good intentions to be misdirected.