Can I modify someone else's pull request and push it back in its branch?

When creating a pull request (PR) on GitHub, you can select the option that maintainers are allowed to commit changes to your PR. You can also do this after the PR is already created: In the detail view of the PR, on the lower right, under "Unsubscribe", "participants". So if you want to commit to a PR of someone else on a GitHub repository you own: Ask them to give you this permission.

From the GitHub help pages: "Only pull request creators can give upstream repository maintainers, or those with push access to the upstream repository, permission to make commits to their pull request's compare branch." (Allowing changes to a pull request branch created from a fork)

Unfortunately, as far as I know, in order for someone to commit to the PR, you would have to give them write access to the repository and can't just selectively give them write access to the PR.

Official GitHub support answer at this date

It would be up to the owner of the forked repository to decide if they want to pull your changes into their fork.

You would modify their Pull Request and merge it into your repository. You could then request them to pull down the changes you have made into his fork which would synch his fork with yours.