Chrome signs me out when closed

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Every time you try to log into them, they take forever because they are old.

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Why am I asking this? I work for a school and they made us built a lab with some old Dell Optiplex 755 computers that were not being used. What I mean by this is, for example, if a user comes and signs into his Google account, or into any website with his own credentials, whenever he closes the browser, the next user will not see him logged in into any of those websites.

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Is there an option on Chrome, an extension or a program that logs you off of everything whenever you close Google Chrome?