Are you allowed to embarrass someone if they have done something wrong?
No, even if somebody has done something wrong, one should not embarrass them, but rather inform them gently of the misdeed (where doing so will be constructive).
The Gemara derives from a verse that even when admonishing somebody, one should not do so publicly to avoid embarrassing him.
The only time when it is permitted to embarrass somebody is in the extreme circumstance of a wicked person where the only way by which others will be saved from his wickedness is through shaming him.
See the Chafetz Chaim’s introduction to Laws of Lashon Hara, Negative Commandment 14, and Be’er Mayim Chaim.