Conversations button in SharePoint Group redirect to Inbox, not to the Group mailbox

Hello Slava_P

Make sure the groups are not hidden from address list:

Set-UnifiedGroup -identity groupname -HiddenFromAddressListsEnabled $false

The group conversation link should work as long as you are a member/owner or have site management permissions assigned via sharepoint for the group. First compare your and other admin's licenses and permissions on the group itself. Next try to open the direct conversation url and see if it gives you any kind of error message , the url should look somewhat like this:   

 https://outlook.office365.com/mail/group/tenantName.onmicrosoft.com/groupAlias/email

Thanks