In Google search, the favicon is not updated.
Installed in the root. Found in Google support that you need to add to the title of the main page of the site
<link rel="shortcut icon" href="/путь/значок.ico">
Tell me where exactly?
As long as you can access "/favicon.ico" there's nothing else you need to do. Just be patient.
As Nathan Malcolm said - as long as the favicon is there, you just need to wait. Generally, Google indexes pages a couple times a week (from my experience owning a smaller website, at least), and re-checks the favicon about once a month, or so. That can differ though, depending on how many visits your website gets and how much Google's crawlers like it (SEO optimization, clickthrough rate, etc).
as Malcolm suggested, if favicon.ico named file exists at the root folder of your forum
then there is no need to add the code in headerinclude template. if icon's file name is
different then favicon related code is required in the template