Pete,
I was looking for a working around.
Also in the category of perhaps too much work (for you, the group moderator) would be to use the Export Group Data function (a tab on the Settings page) to obtain a zip of the group's message content. You could then forward that to the new member.
The downside is having to do it yet again for each new member, as there would (likely) be newer content since your last download.
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Although you stipulated at the start that most members will not log into the website to review messages, that really is the right answer. And maybe you can entice them with the offer of easy-to-access content.
Groups.io makes the login process very easy for email-centric users - they don't have to set anything up, not even a password. Just receive an email message with a link to log in. Hence the helpful link Frances sent you:
/g/GroupManagersForum/wiki/Logging-InShal
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Help:
/static/helpMore Help:
/g/GroupManagersForum/wikiEven More Help: Search button at the top of Messages list