[html] In HTML5, should the main navigation be inside or outside the <header> element?

To expand on what @JoshuaMaddox said, in the MDN Learning Area, under the "Introduction to HTML" section, the Document and website structure sub-section says (bold/emphasis is by me):

Header

Usually a big strip across the top with a big heading and/or logo. This is where the main common information about a website usually stays from one webpage to another.

Navigation bar

Links to the site's main sections; usually represented by menu buttons, links, or tabs. Like the header, this content usually remains consistent from one webpage to another — having an inconsistent navigation on your website will just lead to confused, frustrated users. Many web designers consider the navigation bar to be part of the header rather than a individual component, but that's not a requirement; in fact some also argue that having the two separate is better for accessibility, as screen readers can read the two features better if they are separate.