Google doesn't see the friendly URLs Google doesn't see my SEO friendly URLs, while the other search engines can.

The browsers (IE, Chrome, FF), the RSS, the TweetFeed everything can see the SEO friendly URLs; in the XML sitemap, the system also see them.
Google Webmaster, and Google Analytics also see the friendly URLs, but if I search on my site via Google ("site:mywebsitedomain"), it can see only the node system URLs (e.g. mysite.com/node/6574).

Before the system update, it was working well. My host supports friendly URLs; I have installed the PathAuto, and the Global Redirect modules; clean URLs are enable.

Why doesn't Google see the SEO friendly URLs I set for the site?
Google will only index one version of the same page. It sounds like you have multiple URL's showing the same page. This is called dublicated content. Google will just choose one of the URL's, in your case, the wrong one.

When you enter mysite.com/node/6574 are you redirected to the friendly url? If not, then the GlobalRedirect module is not active or there is no aliases for the URL.

So make sure it's active and in admin/config/system/globalredirect check Deslash and Non-clean to Clean to further minimize dublicated content.

Also make sure that all internal and external links are pointing to your clean url's.

If the above is done, you just need to wait. Be patience - it can take weeks/months.

And forget about the sitemap. The important part is your internal links.