I'm not an expert on this subject, but from what I understand there are generally two ways to do this.
- Have the company establish a subsidiary company in the foreign country and make your husband an employee of that subsidiary
- Your husband establishes his own company in foreign country and operates as an independent contractor.
This interesting article talks about this issue and mentions a few countries that have some other options, although Norway is not mentioned: https://www.shrm.org/resourcesandtools/hr-topics/global-hr/pages/paying-employees-abroad.aspx.
Considering how Norway offers strong employee protections, my guess is that some sort of Norwegian corporation or self-employment setup is required.
Option 1 is a major pain for the employing company. Most companies probably would not bother unless they really value the employee or they were planning for a presence in that country anyway.
Option 2 is a major pain for you. Setting up a company in Norway is likely to be complex and expensive. You'd also have to set up a business to business relationship with the company in the US rather than an employer-employee relationship. Your husband's company would have to be getting a much higher income, since it would have to pay taxes, pay for healthcare, and all the other benefits required by Norwegian law. On the other hand, the former employer wouldn't have to pay taxes and benefits anymore.
An independent contractor means that the company they are working with cannot dictate things like vacation or benefits, but it does mean that they can do work for multiple clients. In fact, I believe that some places require that there be multiple clients to keep companies from cheaply hiring employees as independent contractors, but I have no idea how it works in Norway.
So unless Norway has rules I'm not aware of, something is likely going to have to change. I recommend contacting an expert in Norway to get a better idea of what is possible and the specifics of Norwegian law. I'm also pretty sure there are businesses to help your husband set up his own company if you go that route.
As for taxation, taxes are usually paid to the country of residence almost everywhere. Most countries, Norway included, would not look kindly on undeclared income.
The United States is unusual in that it taxes citizens who don't live or work in the United States. However, since the first 90k of income is usually excluded, there's a tax treaty between the US and Norway, and Norway has higher taxes, it's very unlikely you'd have to pay anything to the IRS. US tax law is aimed mainly at wealthy US citizens who live in low-tax countries. You will have to file a tax return though.