Send email from Google-group-email
As long as you are an "Owner" of said Google Group, you can just add the email as an "Alias" in the Mail.app.

Mail --&gt; Preferences --&gt; Accounts --&gt; Select the account that "owns" the group i.e. "myfirst@email.com" --&gt; Alias --&gt; Edit Aliases


Simply add the email address of the owned Google Group.  In your example "mysecond@email.com".