How will it affect me if a friend I send an Invitation Letter decides to stay permanently in Canada?

There is no law anywhere on Earth that states that you *must* invite anyone to Canada. Your friend is quite able to apply for a tourist visa by herself. If your "friend" applies for a tourist visa, gets a letter of invitation from you, comes over and tries to stay, she will be caught and deported. You may face a fine or even a jail sentence for aiding and abetting visa fraud. So, if I were you, I would not bother.