SP to SP SSO using oidc connect


We are implementing a use case where we need to perform SSO between SPs ( across web application / Desktop app).

The web application is basically designed on OIDC flow and has OKTA widget for SSO.

We have a desktop app which has a link to web app ( SP ) and should be able to perform SSO.

Initial Design -

Desktop app is already federated with IDP using SSO spring boot app that gets launched on local machine and when ever desktop login is require, users are required to perform SSO with web app which in turn will provide a token back to desktop app ( using web socket) and users has valid authentication context.

Since users has valid authentication context, initial design was to pass id token back to web app and performs SSO.

Somehow, that approach is not working and launching browser with id token does not perform SSO as the web app requires many more thing such as code, sid.

I am looking for guidance to implement SSO across SPs, any help much appreciated there.