The "Reply" button at the end of the incoming email from the user.

It is necessary that the user can quickly respond to the message of another user, which he received by e-mail.
When clicking on the "Reply" button in the email, the user must go to the page of correspondence with the sender of the letter.

At the moment in the incoming letter the following code is displayed:
[from] sent you a new private message: <br /> <br /> [message]
<br /> <br /> NOTE: this email was created automatically on the website <br /> 'Сверхчеловечество.рф' - [sitename] under the link '[siteurl]'.

Below you need to place a button or link "Reply".
How to do it?

Don't understand what you're asking. A reply button in an email? A reply button is never added to an email. Just reply using the button as part of your email client. Based off "Emails Sent From" within CB > Configuration > General the from address or reply-to address should typically already be the users email address. If you mean for PMs the reply form is below the message they're reading.

Such a button is in all social network mailings that I know.
When you click on the "Answer" or "Open Messenger" button, the user goes to the sending site, straight to the reply form.

I will give screenshots of examples of emails from:
1. VKontakte
2. Facebook
3. Post from my social network.

When you click on the "Reply" link, the user should immediately go to the site, in the chat window with this user, with the ability to enter a response message.
_____________

In my case (screenshot 3), the user can click only two active links in the email:
1. The name of the sender. (For some reason, the link leads to my profile, and not to the sender's profile).
2. Home page.

- First, the link throws on the authorization form;
- Upon successful authorization, I get to my profile (or to the main page of the site).
- After that, you need to profile and click on the icon of the letter.
- Next you need to enter the letter to write the answer.

I wish it was easier to reply to incoming messages.

Have added a feature ticket to improve the new PM notification.

forge.joomlapolis.com/issues/7410

Most likely will just make the "private message" of "[from] has sent you a new private message." link to the PM. There is no reply button or reply page. Reply is done by using the form below the message while reading the message. Please keep in mind this is private messaging. It is not chat like the examples you've provided.

Thank you, I'm glad the improvement ticket

Alternatively, it is possible that after the message in the e-mail was the inscription:
"Read on the site" or "Answer" or so that you can make a redefinition in the language file,
with reference to the inbox manager, which is located at:
index.php?option=com_comprofiler&view=pluginclass&plugin=pms.mypmspro&action=messages&func=received

I was able to insert such a link in the letter.
I changed the language file with the key "EMAIL_NOTE_AUTOMATIC_GENERATION" for such text:
<br /><br /><p><a href="https://xn--b1aaakaba9b1aeojjo1b6af.xn--p1ai/index.php?option=com_comprofiler&amp;view=pluginclass&amp;plugin=pms.mypmspro&amp;action=messages&amp;func=received">Go to the site to reply

NOTE: this email was created automatically on the website.
'Сверхчеловечество.рф' - [sitename] on the link '[siteurl]'.

Already much better.
But there is a difficulty:
If the user is not authorized on the site, the system simply transfers to the main page and reports in the service window: "You do not have permission to view messages".
The user has to go log in (To do this, he must find the authorization form on the site. It does not appear automatically). After authorization, it goes to the main page of the site, but not to incoming messages.
Where can I set up redirection at login correctly?