We got our permit for a day hike June 23 for a group of 4 of us!! YAY!!! We are all experienced outdoors-man and have done extensive hikes and camping (including 50+ milers). However, we have not really ever done snow hikes. The closest snow hike was my brother and I hiked up Mammoth Mountain when it was closed, but snow still covered the top half of the mountain. We started at Canyon Lodge and followed the chair lift up, then over and up Waterfall and Dry Creek, and finishing by going straight up Cornice Bowl (40 degree angle). We did this with no equipment and in about 3.5 hours (3100' climb). With it being late June, there will most likely still be snow on the mountain. And this is no Mammoth Mountain.

Any advice or preparations we should make?
We were all going to get crampons and trekking poles (converted ski poles). Are ice axes overkill for this time of year? I know they are recommended for May summits. Are crampons also slightly overkill for late June?
We are also planning on starting at 3:30-4:00 am and using the Mount Whitney Trail.
Thanks!