The same bard who was using a Hat of Disguise on her Found Greater Steed is now thinking of picking up Mislead.
While mounted on it, you can make any spell you cast that targets only you also target the mount
that means there is only one spell cast, it just has two targets.
an illusory double of you appears where you are standing. The double lasts for the duration... You can use your action to move your illusory double up to twice your speed and make it gesture, speak, and behave in whatever way you choose.
If the bard casts Mislead on herself while mounted, she can extend the Mislead to the Steed. As one spell with two targets, does the bard need one action from herself to run both illusory doubles, or does she run her own double with one action, and the Steed needs its action to run its own double?