I work in Arcmap 10.3.1.
I have raster dataset - spatial reference 3413
My problem is that I can not send this data on a map.
How do I display the data on the card in the correct projection?
This may be too obvious, but your screenshot indicates that the layer you are trying to display is beneath the basemap in the TOC. Try to move it above the basemap to display it.
Does the raster initially display correctly, before adding the basemap?
Do you get any Geographic Transformation warnings?