Thanks Jim Garrison for pointing me in the correct direction. Here the code, which finally solved the problem for me.
Retrieve the group owner of a file
File originalFile = new File("original.jpg"); // just as an example
GroupPrincipal group = Files.readAttributes(originalFile.toPath(), PosixFileAttributes.class, LinkOption.NOFOLLOW_LINKS).group();
Set the group owner of a file
File targetFile = new File("target.jpg");
Files.getFileAttributeView(targetFile.toPath(), PosixFileAttributeView.class, LinkOption.NOFOLLOW_LINKS).setGroup(group);