Personally,

I'd bill the customer. I'd discuss the situation with the HO and explain that while you don't have all the facts, the other company just didn't quite manage to solve the complete problem.

Then I'd explain that if it were one of my Tech's, I'd like to hear from the customer to know about the issue. If it is a training issue for my Tech, I can't fix what I don't realize is wrong.

Then, I would leave it to the client to contact the company who sent out the other Tech.

If it were my company, when the customer called to explain what happened, I would probably offer to pay some if not all of the expenses of the electrician at least and possibly the plumber, if I felt the customer was being truthful and reasonable in this matter. (I might even ask for the telephone number of the electrician who finally fixed the problem). So that I could verify the facts and improve my Tech's abilities.

If I were the customer, I'd appreciate the time of the electrician who fixed the problem and the advice to call the first contractor.

If I were the first contractor, I'd appreciate the opportunity to improve my business through obviously required training. And I would also try my darndest to take care of the customer who has had such a bad experience.

Just my thoughts....