All emails bouncing from Greenbee account to all Gmail addresses.

[#1]

I'm with normal Plusnet and have no problem emailing anyone on Gmail.  My Aunt was with Greenbee, which Plusnet took over, but they refuse to help her sort this out, so I'm asking on her behalf in here.  Just recently, any email she sends to anyone with a Gmail account is bounced with a rejection message like this:

550-5.7.26 This message does not have authentication information or fails to
550-5.7.26 pass authentication checks. To best protect our users from spam, the
550-5.7.26 message has been blocked.

She recently changed her computer to use a different Email program (well actually a computer shop did it and badly), and I'm wondering if she needs special settings to stop this happening?  Should she be using a different SMTP server for example?  Does she need to use a certain authentication type?

According to this article on Gmail, I think because her email address is
greenbee.net, but she's sending the mail using Plusnet servers, Gmail thinks she's
a spammer. https://support.google.com/mail/answer/81126#authentication 

From the page:

To minimize the chance that your messages are marked as spam, set up these
authentication methods:
Publish an SPF record for your domain. SPF prevents spammers from sending
unauthorized messages that appear to be from your domain.
Turn on DKIM signing for your messages. Receiving servers use DKIM to verify
that the domain owner actually sent the message. Important: Gmail requires a
DKIM key of 1024 bits or longer.
Publish a DMARC record for your domain. DMARC helps senders protect their
domain against email spoofing.
For SPF and DKIM to authenticate a message, the message From: header must
match the sending domain. Messages must pass either the SPF or the DKIM
check to be authenticated.

[#2]

"My Aunt was with Greenbee, which Plusnet took over, but they refuse to help her sort this out..."

Greenbee is not a sub-brand of Plusnet which I recognise.

Googling the name points towards John Lewis Partners - see Greenbee Webmail

Though JLP is provisioned by Plusnet, support for it is through a different channel.  You will need to contact those channels direct, not via the Plusnet support phone numbers or this forum.  See Contacting John Lewis Broadband — John Lewis Broadband

JLP has a support page for setting up email configurations here - Setup guides — John Lewis Broadband - where in it states that the SMTP server should be relay.johnlewisbroadband.com

There is also a relay.greenbee.net SMTP server as well.  I note that Greenbee.net maps on to Plusnet's old mx-ironport servers.

If the JLP setup is anything like that of Plusnet then SPF etc is not supported and arguably not needed.

"This message does not have authentication information" - is she using AUTHENTICATED SMTP - that is connects to SMTP using her user name and password?  What happens if she tries to send email using the Webmail interface as above?

[#3]

Greenbee was taken over by John Lewis which was taken over by Plusnet (which must be true since you say Greenbee maps to Plusnet servers).  If she phones John Lewis as she was told to, she gets a Plusnet person.  She left an angry message with Plusnet complaints who have finally got back to her and told her to test John Lewis webmail which also failed, so presumably it's a server misconfiguration (or Gmail being very fussy about multiple names for the same company, since nobody else bounces her messages).