Hi @Olivia,
I have not found a way to enable session sharing between Edge and IE mode by directly modifying the GPO or registry. However, you can consider modifying the Enterprise Mode site list XML. The Enterprise Mode site list XML is extended to allow more elements to specify session cookies that need to be shared between Microsoft Edge and Internet Explorer.
Sharing example
<site-list version="1">
<shared-cookie domain=".contoso.com" name="cookie1"></shared-cookie>
<shared-cookie host="subdomain.contoso.com" name="cookie3" source-engine="Both"></shared-cookie>
</site-list>
For more detailed information, you can refer to this link:
https://learn.microsoft.com/en-us/deployedge/edge-ie-mode-add-guidance-cookieshare
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Best Regards,
Jinxin Wang