Is a divorce considered to have occurred and a single repudiation (talaq) counted if the husband swore an oath of divorce upon his wife not to do something, then allowed her to do it by informing his mother of his approval, and the wife proceeded with the action after she learned of the approval, even though the husband did not directly inform her of that?

An oath of divorce is governed by the same ruling as a conditional divorce, and it cannot be retracted. So, if the wife does the action sworn upon, the divorce takes place, whether the husband permits it or not. The husband's intention is considered if it was specific, such as him preventing her from doing something at a particular time, or if he prevented her for a reason and that reason ceased to exist.