I offered them my help nevertheless they refused to accept it.
Strictly speaking, this is grammatically incorrect.
You could say: I offered them my help; nevertheless, they refused to accept it.
Or:I offered them my help; they nevertheless refused to accept it. This example is almost the same: It is November however the temperatures are very moderate.
It is November; however, the temperatures are very moderate/It is November. However, the temperatures ...
There is, however, a small difference. However you look at it, the weather is unusually mild.
The two sentences are just two of the ways of using however. Below is the entry for 'however' from the Collins English Dictionary (my go-to source of information on English):
1. adverb You use however when you are adding a comment which is surprising or which contrasts with what has just been said. This was not an easy decision. It is, however, a decision that we feel is dictated by our duty. Some of the food crops failed. However, the cotton did quite well. Higher sales have not helped profits, however. Synonyms: but, nevertheless, still, though   More Synonyms of however 2. adverb [ADVERB adjective/adverb, ADV many/much] You use however before an adjective or adverb to emphasize that the degree or extent of something cannot change a situation. [emphasis] You should always strive to achieve more, however well you have done before. However hard she tried, nothing seemed to work. However famous you are and however rich, losing a baby is the same the world over. However much it hurt, he could do it. Synonyms: no matter how, regardless of how, in whatever way   More Synonyms of however 3. conjunction You use however when you want to say that it makes no difference how something is done. However we adopt healthcare reform, it isn't going to save major amounts of money. Wear your hair however you want. 4. adverb [ADV many/much, ADV adv] You use however in expressions such as or however long it takes and or however many there were to indicate that the figure you have just mentioned may not be accurate. [vagueness] The 20,000 or however many who come to watch would love to be out on the pitch. Wait 30 to 60 minutes or however long it takes. 5. adverb You can use however to ask in an emphatic way how something has happened which you are very surprised about. Some speakers of English think that this form is incorrect and prefer to use 'how ever'. [emphasis] However did you find this place in such weather?

'However' is a word with many uses. However, the use in the examples for the lesson is incorrect.