In such a scenario, sendmail can be configured to masquerade the domain name (From test.com to other.com), effectively.
Before changing the default sendmail configuration, the Sendmail Configuration Files and the M4 macro processor must be installed. M4 reads the sendmail.mc file and produces the sendmail.cf configuration file read by the sendmail application.
Next, we need to modify the sendmail.mc file, located in /etc/mail. We’re mainly concerned with the MASQUERADE_AS, masquerade_envelope and masquerade_entire_domain lines. These lines will be prefixed with delete through newline (dnl) statements. Delete dnl from the beginning of
dnl and FEATURE(masquerade_entire_domain)
Change the MASQUERADE_AS line to reflect the domain name (other.com) that the from address should display (domain of outgoing mail).
Now any user which send the mail with address email@example.com will have firstname.lastname@example.org as a display name going out from this sendmail.