Instinct - does anybody know if software version 6.52 is available for installation on Instinct?

Auto update only updates to the latest STABLE software release.
Beta releases are meant for testing on devices not used as the "daily driver", stay away from them as long as you have to ask the question you asked. When it is considered bug-free, the last beta will be released as the new stable release and push onto your watch via Connect.

Ok, it was not clear for me to see which version is still beta or not. So it looks that 6.10 is still the latest stable version then. 

It is not your fault to understand like that everybody can be confused, even support center and I wanna tell you my experience. I contacted the livesupport for about an hour or so and the supporter on the center even didn't know which one was beta and stable version. I got a lot of instructions for my device on how to update the upper version-6.52 but it was useless . I made this call 2 weeks ago and what I just wanted is to get the release 6.52 and so body battery widget.

ersinmart said:

I contacted the livesupport for about an hour or so ... what I just wanted is to get the release 6.52 and so body battery widget.

You are telling you held the support one hour long just to learn how to install an unofficial beta version? It is explained in the very first "sticky" post on this forum. So instead of offering help to people with true troubles, they are spentding their time in solving similar cases. Don't be then surprised when they are innefficient. 

The only effect such behaviour will have, is that Garmin will stop publishing information about their beta versions, and will keep it just for a closed group of beta testers.

the main problem is that garmin hasnt mentioned this update site - "https://www8.garmin.com/support/download_details.jsp?id=14527"  is a beta update site. This lack of info will eventually increase their workload of support center. I dont care about sticky notes on a forum. even supporter doesnt know that update site is for beta users and she/he is still unaware of sticky note you mentioned

ersinmart said:

the main problem is that garmin hasnt mentioned this update site

On the mentioned page, it is written in a half an inch big font directly in the title of the page, that it is a beta version, and it is mentioned several more times in the text of the page. Besides others directly in the first paragraph, under the title.

ersinmart said:

This lack of info will eventually increase their workload of support center.

It is not the lack of info. It is the lack of respect, and the laziness to read written text, that increases the load on the support center.

ersinmart said:

I dont care about sticky notes on a forum.

Maybe you should. If you are able to find out on the forum that there is a beta version with a new feature, you should have the decency to read the notes and installation instructions. And the sticky posts, of course, too.

Yes, the support is useless. I have the 3rd Instinct watch and I have spent 2x3 weeks without it waiting for exchange, just because the support did not know the "Master reset", which I eventually found on the Garmin web page, but described with a mistake, and without mention that is necessary to wait for 3 minutes.

  I suppose that people in the support are not specialised to one product, or a few products, and so they have only very common information. The best way is to consult users in the forum.

"Instinct software version 6.54 Beta

Notes:

• For any issues that you encounter, please send an email to instinct.beta@garmin.com. In your email please include the software version, and a detailed description of your issue, along with any steps to reproduce. Please note that you may not get a response to the email unless we need more information on the issue you report.
• Although this software is believed to be reliable, it has not yet been released for production and should be used at your own risk."
  
  Well, well...