I've done this a few times
Use an IP intercom system with a wireless IP link from gate to house, all they require is internet at the house and power at the gate, the ubitquiti links work a great distance and are a point to point access point, the contacts to open the gate are in the intercom door station and wire direct to the motor itself.
i recommend Dahua as the brand as they can link into the same app as cctv if they wish to get that installed at later date.
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