iTutor English Updates

Hello! I will repeat my question so that everything is clear. In case of verbs describing removal or putting on of clothes such as "put on/take off/remove/slip on/off over the sweater/head, are they always followed by "over"? Just to illustrate, let's consider the following sentences. 1. He usually puts a jacket on over his sweater in the winter. He usually puts a jacket on over his head in the winter. (It says that he adds a layer to his sweater). (It says how he puts a jacket on - over the head). He usually takes a jacket off in hot summer. He usually takes a jacket off over his head in hot summer. Does it mean that using "over" in item 1 is obligatory? I think so. 2. Last night she slipped a jacket on over her sweater. Last night she slipped a jacket on over her head. (It says that she added a layer to her sweater). (It says how she put a jacket on quickly - over the head). Does it mean that using "over" in sentence 2 is obligatory? I think so. 3. Because of unbearable heat, they removed their heavy woollen shirts over the head/their heads Does it mean that using "over" in sentence 2 is obligatory? I think so. So to sum up. Whether we remove our clothes from our bodies or we put them on / slip them on we do so over our heads. What do you think of my reasoning? Is it true. In case it is not the case, :-D please put me right. Thank you.