See the Gnus Manual node on Startup Files. This mentions the various files and related settings that Gnus uses to save information such as ranges of read/unread articles per group. I believe configuring and syncing these files in addition to your
gnus-init-file will solve your problem. I think the main files are located at
~/.newsrc (cross-newsreader) and
~/.newsrc.eld (Gnus-specific) by default, but I'm not 100% certain.
Gnus in Emacs 25.1 comes with a new feature called the Gnus Cloud, which uses an IMAP server to synchronise your Gnus data across machines. I couldn't find a recent enough version of the Gnus Manual online, so I uploaded a plaintext paste of the relevant nodes as of late March here.
You may also be interested in this EmacsWiki article on syncing Gnus.
Documentation of variables:
For more peace of mind, also read documentation of variables: