Appointy provides two-way sync with Google Calendar, any appointment booked with Appointy will automatically get synced with Google with all details (Service name, client details).
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Now what you need to do is integrate Appointy and Google calendar on same page of your website.
By doing this, both the calendars (Appointy and Google) will be visible to your clients on the same page. They will see the client details on Google calendar and can book an appointment on Appointy calendar.