What is CURL equivalent for GenieACS "SUMMON" button?

…or what is “SUMMON” button do behind scene?

Second question:
I trying to get actual SSID from router but when device connect to GenieACS for the first time SSID value is not cached so I tried (I guess) almost all avaialble API calls but nothing works only “SUMMON” button from GenieACS.

How will looks command for ex. in CURL to get and cache actual value of router SSID GenieACS ?

From the API:

curl -i 'http://localhost:7557/devices/202BC1-BM632w-0000000/tasks?connection_request' \
--data '{"name": "refreshObject", "objectName": ""}'

Answer to your second question available here:

curl -i 'http://localhost:7557/devices/00236a-SR552n-SR552NA084%252D0003269/tasks?timeout=3000&connection_request' \
--data '{"name": "refreshObject", "objectName": "InternetGatewayDevice.WANDevice.1.WANConnectionDevice"}'

~ R