Answers:
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Method 1
I’ am not sure how are you handling the repeater in your code behind, I used this repeater and the background did change while hovering the links:
<asp:Repeater runat="server" ID="rpt1">
<ItemTemplate>
<li>
<a href="#" rel="nofollow noreferrer noopener" class="mySprite id<%# ((int)DataBinder.Eval(Container,"ItemIndex")) % 6 + 1%>">
<font color="white"><%#Eval("Name")%></font><div> </div></a>
</li>
</ItemTemplate>
</asp:Repeater>
PS: I used the top most “CSS and Mark up” you posted and tested it on IE.
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