Use built-in task authorized_key to install keys for sysadmin.
Use built-in authorized_key task in the playbook:
	modified:   roles/make_servers_sshable_by_sysadmin/tasks/main.yml

make the operation to make remote hosts able to ssh onto each other into
one script:
        deleted:    scripts/README.md
        deleted:    scripts/collect_hosts_pub_keys.sh
        new file:   scripts/make_servers_sshable_on_each_other.sh