。
Mother likes to go out wearing this coat.
Literal
Mother [topic-は] this coat [obj-を] wear and go-out [nominalizer-の] [subject-が] is-liked.
好き takes the thing liked with が, not を — one of the canonical が-takes-the-object verbs (好き, 嫌い, 上手, 下手, できる, わかる, ほしい, ...). Here the 'thing liked' is a nominalized verb phrase 着て出かけるの ('the-wearing-and-going-out'), which のが smoothly turns into a が-marked subject. The て-form on 着る chains it to 出かける as a manner/state description of the going-out.