Q1. Does my friend is required by Law that he should inform the Home Office that his wife left him with another man.
Q2. Can my friend ask the Home Office to cancel her ILR and send her back to her home country.
For Q1, Home Office only need to know if the application which was made for ILR in the first place was as a result of fraud. This could mean that the spouse forged the signature in order to get ILR, or, spouse was already married but did not declare this. Otherwise if everything was legal and clean and there were no illegal operation then the Home Office cannot do anything and there is no need to persuade them to cancel ILR because it is against human rights unfortunately for the victim.
For Q2, there is no way ILR can be cancelled because it becomes law once the legal requirements and consent of the husband to accept ILR for wife was made in the first place. This should have been taken into consideration before applying for ILR. The husband could have delayed ILR and see if the wife changes behaviour even though she would be living illegally as her spouse visa has expired.
However, the only way to cancel ILR is this:
a. She is a threat to national security
b. She goes abroad for more than 2 years and then when she returns to UK, her ILR automatically becomes invalid.
Otherwise there is nothing you can do. She will of course claim for naturalisation and a british passport. This is like breaking your heart because she betrayed. But in life you have to move on and pretend she never existed and that way you will be wiser and warn others and be successful with your future.
If you let this matter drag on, only your life will suffer. Thats my advice so don't try to take this to heart just move on like a brave soldier. Just leave everything to Judgement Day where those who did wrong shall be punished.
Hope that helped.