ungrounded switched outlet with three 2-wire cables coming into it -- make GFCI

[home_electric]

I'm just a DYI-er here and very new. I have an ungrounded outlet that is completely controlled (it's not half-controlled) by a wall switch. The place I'm in was built in 1957 and this outlet appears to have a jumper wire from neutral to ground. Notice in the attached pics that there are 3 2-wire cables coming into this receptacle.

I'd like to install a GFCI in its place, but I am at a loss as to how to wire the GFCI with what I have going on in this scenario. Attached are pictures of the wiring in this outlet.
THANKS!

 

[Marvo]

Maybe it was mistakenly wired that way or maybe the grounding arrangement is TNC which isn't very safe and isn't used nowadays.

Is that outlet the only ungrounded one or do other outlets and switches in the premises also have a link between neutral and earth terminals?

Either way there's no way you can leave it like that and a GFCI won't work with that wiring.

 

[home_electric]

The few others on the same circuit that this one is on tested as open ground. I also wish this one wasn't controlled by wall switch as I want it to always be on. I wasn't sure if I could simplify this is any way so that I could eventually replace it with a (ungrounded) GFCI receptacle.

 

[home_electric]

Maybe it was mistakenly wired that way or maybe the grounding arrangement is TNC which isn't very safe and isn't used nowadays.

Is that outlet the only ungrounded one or do other outlets and switches in the premises also have a link between neutral and earth terminals?

Either way there's no way you can leave it like that and a GFCI won't work with that wiring.

The few others on the same circuit that this one is on tested as open ground. I also wish this one wasn't controlled by wall switch as I want it to always be on. I wasn't sure if I could simplify this is any way so that I could eventually replace it with a (ungrounded) GFCI receptacle.