How can the scheduler of Task.Run be changed?

I need to run a few tasks that require to be ran using a specific scheduler as they would fail otherwise since the objects they do instantiate can only be from a specific thread.

Using Task.Factory.StartNew works well, the only thing is that the syntax is a bit cumbersome.

So I came up with the idea of writing an extension method that could allow me to keep the terse syntax of Task.Run but specify another scheduler than TaskScheduler.Default. I am having a hard time figuring out how to write such extension method, however.

Question:

How can I change the scheduler of Task.Run it is meant to run with, if possible at all?

Code example:

using System;
using System.Threading;
using System.Threading.Tasks;
using UnityEngine;

public class NewBehaviourScript : MonoBehaviour
{
    private async void Test()
    {
        // game objects can only be created from Unity main thread, get its scheduler

        var scheduler = TaskScheduler.FromCurrentSynchronizationContext();

        // 1. syntax using factory, works but is a bit cumbersome

        await Task.Factory.StartNew(
            () => new GameObject("test"),
            CancellationToken.None,
            TaskCreationOptions.None,
            scheduler
        );

        // 2. ideal syntax though it will fail since it'll run with the wrong scheduler

        await Task.Run(() => new GameObject("test"));

        // 3. ideal syntax but how to implement .Schedule method?

        await Task.Run(() => new GameObject("test")).Schedule(scheduler);
    }
}

Extension method:

public static class Extensions
{
    public static Task<T> Schedule<T>(this Task<T> task, TaskScheduler scheduler)
    {
        // how, if possible at all to write this method?

        throw new NotImplementedException();
    }
}