Ory is the largest open source community in the world for cloud software application security. We maintain advanced open source security software solving authentication, authorization, access control, application network security, and delegation. Ory implements a variety of industry and best-practice standards including OAuth 2.0 / OAuth 2.1, OpenID Connect, Zero Trust Networking, Google Zanzibar Policy Framework, FIDO2 U2F, WebAuthn, TOTP, and more.

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is there any way to change user password through admin API ?

no, but you can trigger recovery through the admin API and send the link to the user
that way a password is never leaked

is there any difference between changing password through public API and admin API except we don't have to init GET /recovery and send a POST to /recovery endpoint?

what do you mean by "change password through admin"? you have to initialize a recovery flow and complete it over the public API in both cases, right?