What admin access does connecting an integration need?

It differs per integration.

Each person connects their own Google Calendar, Office 365 (Outlook) and Slack from My Integrations, while Tempo, Microsoft 365, Google Workspace (beta) and Slack can be configured once for everyone by a Jira admin under Administration > Global Integrations; Slack additionally needs a Slack workspace administrator to install the app into the workspace.

Permissions and Access Requirements - where this is covered in full.
User Permissions - more questions on this topic.

What admin access does connecting an integration need?

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