I had an older family member who fell victim to a very similar scam. Once she installed their remote software, they connected and used her browser to transfer a large sum, from one of her bank accounts into her main bank account.

They then convinced her that they had "deposited" the money into her account by mistake. She could clearly see the extra money in her main account and so believed them. The person pleaded with her to return the money via transfer, and keep it secret, or he would lose his job.

Being a good person, she wanted to help him and not cause a problem.

Fortunately, the teller at the bank recognized it as a scam, but it was a close call.